Roger Penrose – Is Mathematics Invented or Discovered? [video] | Hacker News Reader

Roger Penrose – Is Mathematics Invented or Discovered? [video]

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Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0]

In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory.

That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1]

[0] https://en.wikipedia.org/wiki/Intuitionism

[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...

> Applying intuitionist mathematics to physics we can come to the conclusion that time flows

Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.

>Can you expound on this a little more?

Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough.

I am sure other HN users understand this stuff better.

From the wikipedia source: In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?

The mathematical definition of intuition is not the colloquial definition. For example, in intuitionistic mathematics, you can't proof by contradiction, which means that some things that are "intuitively true" in the colloquial sense, are inaccessible.

The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic.

But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.

I think in your first paragraph you're confusing intuitionistic logic and linear logic --- intuitionistic logic still lets you "clone" and "delete" propositions freely.

The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".

When you separate internal and external logic, for example in topos theory, the external logic is still classical.

For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra.

However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it.

The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.

I think you are right about the comment about linear logic. I must be confusing A U A with A U ^A.

I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.

It's the other way around, classical logic is a restriction of intuitionistic logic that arises when you affirm the Law of Excluded Middle.

I think you're both right in a certain sense. The set of provable statements in intuitionist logic is a subset of what is provable in classical logic, which is what I believe the parent was referring to.

You're certainly right that any constructively provable statement is classically provable, that follows from intuitionism/constructivism generalizing classical logic. But I don't see how we could interpret the following to mean that.

> you can do intuitionist mathematics in classical mathematics

> But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around.

There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.

You shouldn't be mapping propositions, you should be mapping theories. See my comment below on the internal logic of a topos.

What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.

Why do you think the same thing doesn't apply to theories? Since you mention topos theory, the double-negation topology is actually a basic construction in topos theory, used to construct a dense Boolean subtopos of a given topos.

You can always generate substructures with closure properties. The subobject structure comes from all the subobjects—so it is the default logic of the topos. Of course you can do the same thing here as you mentioned since it is a Heyting algebra at the least.

My point is something else: I don't know of a way to have a intuitionistic exterior logic and a classical internal logic.

I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry.

Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.

Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.

> on this planet

Location was not specified ...

That's only true if you limit this to conscious thought. Lots of organisms perform impressively complex calculations. The desert ant odometer is one classic example:

https://jeb.biologists.org/content/210/2/198

I've always figured it's our visions ability to perceive wholes, one of something. A clear conceptual border between two entities.

Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).

Agreed, I wonder if the same amount of abstract concept would be possible though, can you feel a graph? could you do trigonometry with just hearing and sense of touch?

"could you do trigonometry with just hearing and sense of touch?"

Can you get data on this from a Monte Carlo method of study on match play between sighted and blind players playing go and chess at distance and blind which requires a mental model of the board state and translations?

"two of our own species [discovered calculus]"..? Which two species are those? I assume by "own own" you mean Terran species, and one of them is Homo Sapiens, which is the other one? Really curious...

Leibniz and Newton. Two [members] of our own species.

And Archimedes makes three.

Take your stinking paws off me you damn dirty ape!

Probably Homo Econimus. If Homo Econimus is good at one thing, it's math, to the exclusion of pretty much everything else.

Do Dogs Know Calculus?

http://www.math.pitt.edu/~bard/bardware/classes/0220/dkc.pdf

And yet, most blind mathematicians are geometers, which suggests your theory is incorrect.

I'm not convinced that blind mathematicians count as their own species, nor that they echolocate, and so it doesn't seem to say much about his theory one way or the other.

> nor that they echolocate

How can you be sure?

Humans can learn/be taught echolocation. https://en.wikipedia.org/wiki/Human_echolocation

We could perhaps modify the theory, thinking of the mechanics of research. Progress is made by an individual in the niches where others have not managed yet. Therefore the advantage in geometry for a blind mathematician may be in a different mental model of spaces.

It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]

recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]

This is a quote from a book by Alexei Sossinsky, quoted here: https://onionesquereality.wordpress.com/2011/07/31/blind-geo...

Ah! That's where I read it! Thanks, I read that book a year ago and couldn't remember where I'd heard that factoid.

Maybe vision is helpful with elementary geometry. I don't know how vision would help with differential geometry, geometry of affine spaces, topology, algebraic geometry.

Maybe visualization is more helpful, but you can visualize things without seeing.

A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required.

From another point of view, math theories are congruent, and you can explain the same things without using geometry, but using algebra, or differential equations or number theory and so on. You certainly can explain any math concept using just modern set theory - maybe it is not the most convenient thing to do.

Category theory is thoroughly visual, though.

A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry.

Except once axioms have been fixed, the resulting geometry will be the same wether you perceive reality via visible light or echolocation.

I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode)

https://corecursive.com/035-bartosz-milewski-category-theory...

And really one of the biggest pieces of proof of this is that some of the earliest mathematics developed was planar geometry.

But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world.

This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.

But, this makes sense because we really don't ask the same question about human language. We almost never ask: is human language constructed or discovered?

>This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.

As I see it, mathematics is both discovered and invented.

We can model every existing thing in every possible world using math. Even if both the set of all things that might exist and all possible mathematical constructions are infinite, the later is larger. That's because we can also construct mathematical models of things that doesn't exist.

So it looks that from the set of all possible mathematical constructions, we extracted a subset that maps to objects in reality. That looks like a discovery process.

But we also constructed mathematical models of things that don't have corespondents in reality, so that much be more of an invention process.

Let's pretend for a bit that we forget all what we know and tomorrow we will start inventing things again. Or that a species of aliens start fresh on a planet.

The mathematic theories and notions we and the aliens might discover, build or invent might be different than the theories and notions we know today but would probably be equivalent. That suggests that mathematics just exists somewhere in its own world waiting to be discovered.

I too, subscribe to this view. In my mind, reality is nothing more than a vast integer variable field with interactions between some variables. All our mental models, from vision to math are attempts at approximating that complexity down to a level that our brain's limited processing capacity can handle...

Why integers one might ask. For me, even rational numbers are such an approximation.

This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge.

It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.

The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality is mediated by our brain's own idiosyncrasies and limitations of thinking.

It's no more surprising that external objective reality is mathematically predictable and describable than it is that our eyes can see shapes and some, but not all, light. That's the purpose of eyes and they tell us enough of what we need to know that we can survive.

Our brains are exactly the same thing- they tell us a story in a way which helps us to survive. Actual reality may be beyond our capacity to conceive of or worse, may seem like nonsense to us because it's aggressively illogical or specifically contradictory and reality therefore makes no "sense" to us.

>It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.

But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)?

I would say our brains do have limits. I can't well envision a 4D object. But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist. Even if I was a simulation that didn't even have a brain, much less eyes, the concepts I come up with would exist more than the flesh I incorrectly thought I had.

>>But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist.

I agree.

But this assumes logic and at least the principle of non-contradiction: not both A and not A - is how reality "really" is.

We can't get past the idea that it must be this way because only provable nonsense lies on the other side of this assumption. But our brains may be fundamentally unable to process ultimate reality and the nonsense a contradiction represents may be a statement not about reality but our brains, our thinking.

So logical contradictions aren't actually nonsense, they're the sound of us hitting the walls of what our minds can conceive of. All animal have such limits. We assume those limits look like darkness- stuff we can't peer into. What if they look like impossibility instead?

That's the point of view I'm entertaining here. I am not saying this is true. It certainly isn't useful or provable as far as I know, but it is possible.

The practical value of such an exercise, if it has any (and I think it does) is to twofold.

One seems to expand my imagination to the maximum extent pops me out of the assumptions that frame my thinking and this seeps into my thinking about things, technical problems, generally.

Two it confers humility and a certain openess and makes me less judgmental. So that, for example, when I hear or read people with claims to spiritual knowledge I don't automatically blow them off as crazy / bitter / ignorant because what they're saying "makes no sense".

Thinking and talking about Ultimate Reality capital U capital R, ought to fill us all with humility if we're being intellectually honest by our own standards. Yet, I find people totally lack that humility. They make huge pronouncements about Ultimate Reality which they can't really be sure of, and the effect this has on the world, and how we think of each other, and therefore how we treat each other, and even the effect on one's own mind, is one of diminishment generally.

Mea culpa, I was one of those people and I didn't like it.

If you get down to the core of knowledge and how we know something, this is what's really there, and it's good to be reminded of it.

Not exactly the same but 2-dimensional beings on a curved surface can discover the 3rd dimension using math, we did this proof in a differential geometry class.

I would love to see a proof of it if you happen to have it!

https://fsw01.bcc.cuny.edu/luis.fernandez01/web/texts/dgcs.p...

Ctrl-F for "2D" (2-dimensional creatures), although I can't find the "discovery", maybe I remember it wrong.

Btw, this was one of the mindfucks I learned studying physics. Other ones are:

  - dimensional analysis: checking if formulas can be wrong, and more importantly, finding formulas (eg, period of a pendulum) just by analyzing units that can be involved (no constants obv)  
  - similarly, rate of growths in dimensions: why giants can't exist (bone resistance is proportional to section area and load (weight) is proportional to volume  
  - Conservation laws derive easily from symmetries (energy from invariance in time, momentum from invariance in direction etc)

This is brilliant, thank you!

I think you may be talking about the curvature of the surface (the Riemann tensor). It is intrinsic, i.e. does not depend on the notion of the surface being embedded into a higher-dimensional space.

Assuming a species is intelligent enough to discover math, if they have some significantly different way of experiencing the world I think they would take a different path but would eventually reach the same destination we are heading towards, likely merging with us by the point we are at now.

I could see the layman understanding of geometry being quite different than ours, limited by their own visual system. But their mathematicians and our mathematicians would discover the same math, through different paths and perhaps with different strengths and weaknesses and a different path of educating budding mathematicians from the grade school level to the post doc level.

Tell me more about the difference between "potential infinity" and "actual infinity".

Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?

I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.

- Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3]

This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.

Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.

[1] https://www.quantamagazine.org/landmark-computer-science-pro...

[2] https://www.quantamagazine.org/does-time-really-flow-new-clu...

[3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...

The video timestamp you linked does not say "free will exists". Is says "humans has free will implies element particles have free will".

His definition of free will is quite limited. He only suggests that free will is limited to a "free" choice between options within one's power.

Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options.

https://youtu.be/tmx2tpcdKZY?t=1074

https://en.wikipedia.org/wiki/Free_will_theorem

Free will is a philosophical concept which is essentially dualist. It's an attempt to reconcile two subjective experiences - the feeling of making a decision, and the fact that sometimes we can't predict the actions of others - with universal determinism.

There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will with weather forecasting.

Even if that weren't true you still can't get to Conway's Step 3, because "free will" could just be statistical noise, and not willed in any sense at all.

>Free will is a philosophical concept which is essentially dualist.

Is it though? I came to the conclusion that one could consider free will the self-realization of a person in space-time -- in other words, free will is the only and only choice one can make (at each decision they face), that they make because of who they are thus far.

In that sense, "free will" is the same as the "person" (or the person's essense) -- and would it make sense for it to be any other way? Randomness wouldn't be free will, and having a second entity (e.g. a soul in dualism) do the decision making just moves the question and degree upwards (from "how the person has free will?" to "how the person's soul has free will?")

If a person faced with a choice A or B could go either way, then I wouldn't call that really free will (even though for some reason that's what most people have in mind when talking about the subject).

Either a person is a concrete personality/being and that manifests through its choices (which means that when faced with the A or B dillema they can only chose one or the other based on who they are), or they have some degree of randomness in their decision making (which makes them less of a person in my eyes -- randomness is not tied to our person, it is, as the name reveals, random).

When some people say "free will" they actually mean "free (independent) from the person". But that's not a will then -- a will denotes something being tied to a person (a person is their will).

You're conflating different kinds of randomness here.

A coin flip and a hand of cards are both random, but obviously cards are 'more random'.

This can be quantified - 'more random' is information complexity, or information entropy. (Two names for the same thing.)

If free will exists, then it must necessarily be a singularity of infinite information complexity, which is the same thing as a singularity of infinite randomness.

> Free will is a philosophical concept which is essentially dualist.

It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.

First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement. Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it. Pulling a quick quote, the Compatibilist position can typically expressed by the following:

Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.

This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.

Therefore, while I will not support a metaphysical dualist basis for ‘free will’ it is clearly incorrect to dismiss an argument of such based on 1) an appeal to authority which is unsupportable (in that your claim about the position of a majority of ‘experts’ seems unsupported), and 2) that even if it was supported, that said appeal to authority is a valid refutation of an argument. Further, dismissal of an argument by choosing to substitute a contested definitional term (the meaning of free will) for one that is clearly not the same is not rhetorically valid.

> First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement

Almost 60% Compatibilist, the remainder evenly split among three other options: https://philpapers.org/surveys/results.pl

> Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.

You're assuming this is relevant. Turns out, it's not.

> Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it.

Nobody definitively understands free will. Some people conjecture it has certain properties, mainly incompatibilists. So far they have mostly been wrong.

> This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.

Experimental philosohy suggests pretty definitively that people's moral reasoning agrees with Compatibilism: https://www.researchgate.net/publication/274892120_Why_Compa...

> that even if it was supported, that said appeal to authority is a valid refutation of an argument.

The other poster provided no argument, they just made an unsupported claim.

Potential infinities are just saying, you don’t have an infinity in your hand — you just have a recipe for arbitrarily large numbers (or arbitrary accuracy on real numbers, etc).

Eg, the “natural numbers” are a recipe to make as many as you want (by taking the successor of the last one) but not an actual (as in existing) infinite collection — the only ones which exist are the ones you construct, and the “potential infinity” refers to the fact that you can always make a new one.

"Actual infinity" is merely an abstraction, not unlike many others. It is very useful. Abstract thinking is part of what separates humans from (other) animals.

While there is technically no proof, there is a lot of evidence that the universe is infinite. In particular it's curvature is flat within the margin of measurement error which implies an infinite universe outside the observable universe.

> Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0]

The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.

I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.

I was merely pointing out that the "mental" dependency of intuitionism isn't strictly necessary. Take it out, and you're left with an equally valid constructive logic.

The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are physical. So it's not surprising that the structure of the physical world should be reflected back on itself in the structure of mental processes. It's universal Turing machines all the way down.

So, if there is a difference between invention and discovery, then dualism is true?

Yes, though that's a bit of an odd way of putting it. It's kind of like saying that if matter can't be broken down into smaller constituents beyond a certain point then atomism must be true. It's more of a definition of dualism/atomism than a logical entailment.

One man's modus ponens is another man's modus tollens.

> Invention is a kind of discovery.

Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)

> human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery

Why not? It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember. All of these are (on the assumption that dualism is false) physical processes.

> It's a discovery of visual features that the human brain and unaided eye can easily discern, and the discovery of a correspondence of those visual features to concepts that the human brain can easily remember.

Hm. Yes, I suppose you could look at it this way. However, there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves. All humans discover the same mathematical structures, but different humans invent different mathematical notations to describe those same mathematical structures.

So I guess it's a matter of whether you want to focus on the similarities or the differences between those two cases.

> there is still an element of arbitrariness or choice involved that is not present in the discovery of mathematical structures themselves

That's not true. There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not. And math is chock-full of this kind of thing. Even geometry comes in different flavors: Euclidean and non-Euclidean. And we don't even know which of them corresponds to physical reality on cosmological scales!

> There is this thing in math called the "axiom of choice" which you can, quite literally, choose to accept or not.

No, there are these different mathematical structures: set theory with the axiom of choice, set theory with the negation of the axiom of choice, and set theory with neither. (More precisely, "Zermelo-Frankel set theory", since there are other set theories.) All of those structures were discovered. And, as I said, different humans invented different notations to describe these different structures.

Humans choose which of these structures to use for particular applications, yes. I don't think that process is either invention or discovery. Not everything humans do has to be an invention or a discovery.

Similar remarks apply to geometry.

I think notation can certainly be regarded as a kind of discovery. As time goes on, notation improves dramatically as a tool of thought. This improvement is a result of feedback over what works and what doesn't. The feedback is discovery.

Invention = discovery in "solution space".

I don't understand why it is useful to argue "invention is discovery is invention". They are fairly well defined in the dictionary and it's a useful distinction to humans. Sure there is a gray area of a wide swathe of overlap, but in general they are useful as commonly understood.

> The apparent distinction arises from our separation of physical and mental processes.

I'm not sure I understand this either. I think our ordinary intuitions would say that both invention and discovery are mental processes. They're just different kinds of mental processes: invention has an element of arbitrariness or choice in it that discovery does not.

I have no problem with saying that mental processes are physical processes; I just don't think that means invention is a kind of discovery.

> invention has an element of arbitrariness or choice in it that discovery does not

Yes, I can understand how someone could reach this conclusion. I used to believe it myself. But if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. Take the airplane for example. Was it invented or discovered? Is there a relevant distinction between the "invention" of the airplane and the "discovery" of the principles of aerodynamics? What is it?

Even the arts can be seen as a process of discovery, specifically, the discovery of what processes and artifacts produce favorable reactions in human brains.

It's really a matter of perspective, of what dramatic narrative you choose to apply. Any story of invention can be re-cast as a story of discovery.

> if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion.

I don't think that's the illusion. I think the illusion is that something as complex, for example, as "the airplane" has to be pigeonholed as either an invention or a discovery, instead of being a complicated process in which both invention and discovery were involved.

> Any story of invention can be re-cast as a story of discovery.

I think this depends on how far you are willing to stretch the meaning of the word "discovery".

No, it's an observation that there is no sharp line between invention and discovery. If you want to dispute that, you'll have to tell me how I can distinguish between the two.

> it's an observation that there is no sharp line between invention and discovery

This is true of pretty much any human categories, yes. That does not mean the categories aren't valid or that they should all be collapsed into one.

Fair enough. But notwithstanding, I contend that my claim is not vacuous. I would say that this particular failure of categorization actually reveals a deep truth, namely, that computation is universal.

> I would say that this particular failure of categorization actually reveals a deep truth, namely, that computation is universal.

I agree that computation is universal, but I don't see drawing a distinction between discovery and invention as denying that fact. In "computation is universal" language, the distinction is just the observation that there are different kinds of computations--moreover, there are different kinds of computations in the category "human mental processes". We don't need to say they're all the same to recognize that they're all computations at bottom.

> "arbitrariness or choice" is an illusion

after reading a little about quantum mechanics I'm not so sure any more.

>Even the arts can be seen as a process of discovery, specifically, the discovery of what processes and artifacts produce favorable reactions in human brains.

That assumes that the artists test their work against real audience and adjust their style accordingly. Maybe they do, but I'm sure that there are artists out there who just create art.

>> "arbitrariness or choice" is an illusion

>after reading a little about quantum mechanics I'm not so sure any more.

Even Brownian motion would suffice, a chemical phenomena we've known about for millennia.[0] The jittering of sodium and potassium ions between neurons can be enough to trigger spike firing in a 'random' way[1]. At the end of the day, we are just complicated salty sacks of stochastic processes trying to reproduce, not much more.

[0] yes, it's 'quantum mechanical' in cause, like all things are. I'd say it's more statistical mechanics than quantum though, but to each their own: https://en.wikipedia.org/wiki/Brownian_motion

[1] https://en.wikipedia.org/wiki/Principles_of_Neural_Science

>The apparent distinction arises from our separation of physical and mental processes.

You make the distinction based on (physical or mental) processes involved in the discovery/invention. I wouldn't care so much about how the discovery/invention came to be and look more into what the discovery/invention is about.

Usually, you can discover things which are present in the real world. Mathematicians are rarely interested in the real world. They invent things which are true (or false) regardless of the physical world.

But mathematicians are present in the real world, so you can choose to see math as the discovery of the behavior of the brains of mathematicians.

lol, yes, you discover how mathematicians think, but what they think is still an invention.

In the end, I agree that if we expand the definition of the universe beyond physical world to contain "everything" then yes, every invention is also a discovery.

> what they think is still an invention

Is there anything that any human thinks (other than straight-out mimicry) that is not an invention?

If I type:

squirgle florb snozbar

would you say that I have invented something? Because if the answer to that is yes, then I'll concede the point (and observe that "invention" is not a particularly interesting concept). But if the answer is no, then I challenge you to draw a distinction between what I just did and what mathematicians do in a principled way.

neither inventions nor discoveries have to be interesting.

>squirgle florb snozbar

yes, you invented something (i think you can even claim copyright on that). I agree that it is not very interesting. mathematicians usually/sometimes produce more interesing inventions.

> yes, you invented something

OK, I guess we'll have to agree to disagree then. I would not call that an invention (and I think most people would agree).

OK let me invent something:

A fignaft is a small pink bird from the island of slamp. It has a short triangular beak well adapted to plucking seeds out of the hard cones of the humnit bush. It was nearly hunted to extinction by the umaglots, until their chief shaman declared the bird to be a repository of the souls of men lost at sea. Since then it has made a remarkable recovery.

In contrast to the mental process required to invent an imaginary bird such as this, the folks who discovered DNA used tools to image cell internals such that the structure of the contents was revealed.

When DNA was discovered, something about the structure of the external world was made clear to us, and many other processes could be clarified as a result. The fignaft bird, on the other hand, only reveals the contents of my mind, and is constructed entirely of existing knowledge.

OK let me invent something:

gnoiwiof wefoin wf oinwfe oinfowin wfeoi fwenoifwe

Do you count that as in invention? If not, what distinguishes it from your "invention"?

You seem to be protesting that the bird invention is nonsense, and no more meaningful than random letters. Regardless of the answer to your question, I think you might be overlooking a different point.

Here is my invention: a new piece of jewelry, a double ring, which you wear on both a finger and a toe at the same time. This is a realizable physical object that (presumably) has never existed before. Is it a discovery?

This double ring may be the instrument that discovers a market for itself, or a visual/tactile aid to remind oneself of something, but in itself this invention is not a discovery of anything that can be said to have existed.

Your comment, made by alanbernstein 3 hours ago on this website hackernews on 4/17/2020 has never existed before. Is it an invention ?

If I must choose between invention and discovery, then yes.

All you did was discover that if you typed those words on your computer, they would appear on that website on that day.

So each distinct event can be discovered, regardless of how predictable it is based on previous events? Each sunrise is a discovery?

"predictability" is only a function of your memory capacity and/or knowledge.

To a scientist/saint who has done a lifetime of study/penance on the past, present and future, every invention is predictable. To a goldfish with 5-second memory, every sunrise, why every breath, is a discovery.

Look at it another way. A scientist 'discovers' the atom. He didn't 'invent' it, right? Now, how did he discover it ? Using a microscope that another scientist 'invented'. So far so good?

Now go down the chain to that scientist. How did he 'invent' the microscope ? Did he make it out of thin air ? No, he was one day playing with 2 optic lenses that another scientist 'invented', and when placed in a line, he 'discovered' that they magnified it 100 times what one lens did. And he told everyone that 2 lenses together is called a microscope that he 'invented'. So far, so good ?

Now, go down the chain to that scientist. How did he 'invent' the optic lens ? Did he create it out of thin air ? No, he was one day playing with glass which another scientist 'invented', and when looking through it, he 'discovered' that they magnified it things 10 times their size. And he told everyone that looking through a glass lens is a 'magnifying glass' that he 'invented'. So far, so good ?

You see the pattern here ? Every permutation combination of nature exists. In other words, all possible inventions already exist waiting to be discovered by humans, or all possible discoveries are waiting to be invented.

The 2 words are synonyms.

Something 'new' that you 'discovered' was someone else's 'invention'. And once you go beyond the deepest chain that your senses can reach, even the physical laws that govern our universe that were 'discovered' are an 'invention' of God. Where did God discover it from ? We can ask Him.

I wonder if there are languages where this question doesn't make much sense, because, say, both invention and discovery translate to the same word just meaning "creation".

No need to wonder. English is that language.

Tomorrow I can come up with a new word called 'oogabookga' which means invention/discovery/creation, and then make enough people accept it so it makes it into the dictionary, and then 20 years later, 2 other folks like us will argue about the semantics between oogabookga and invention and discovery.

That's exactly what happened with invention/discovery 20 (or 200) years ago, and we are those 2 people now.

I think a useful dichotomy is between whether an intellectual product's existence is possible without the existence of the person who created it. Copyrightable works typically could not exist without their creator. Only Prince was capable of creating When Doves Cry and only Jane Austen was capable of writing Pride and Prejudice. But inventions share with mathematical discoveries that they can (and often are) created or discovered separately by more than one person.

You say

> mental processes are computational processes

as if this were obvious but... If you accept the possibility of random events (something deeply related to Quantum Mechanics)... There can be lots of non-computable things out there in our minds...

Such as? "Computational processes" ≈ "arbitrarily approximable by a classical computer," and QM absolutely fits in there: I just solve Schrodinger's equation for the Hamiltonian you care about to whatever precision you need. It will take exponential time, but that seems perfectly reasonable.

Exactly. Furthermore, there is no evidence that quantum processes are at all salient to what the brain does. There is nothing in the I/O behavior of a human brain that provides any evidence of quantum processes going on. There are also good theoretical reasons to believe that quantum mechanic is not salient to the operation of the human brain. The brain is thoroughly thermalized, so it decoheres in time scales that are vastly faster than any of its interesting behaviors.

> The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery.

this is self-contradictory. if invention is a kind of discovery, then they aren't equivalent as you say.

invention and discovery are clearly different but related concepts. hence the question. i didn't invent the sun. i discovered it, for myself, the first time i saw it. this is a structure that exists.

the mathematical question boils down to whether the structures that we find are discovered or conjured up. our physical inventions do discover new things, but these are things that existed whether the invention happened or not. the invention is simply a process.

The words "invention" and "discovery" are also both human conceits. It seems perfectly appropriate, then, for them to distinguish two separate other human conceits.

> The apparent distinction arises from our separation of physical and mental processes.

The distinction is that discovery is of something that exists without human intervention and invention exists because of human intervention.

For example, telescopes are invented ( human made ). The moons of Saturn are discovered ( not human made ).

You created a straw man and argued about dualism. The question of discovery and invention has nothing to do with mind/body problem.

> Invention is a kind of discovery.

Presumably you would agree that not all of discovery is also invention. So doesn't that restore the original question of asking whether mathematics is an invention or discovery which is not invention?

Yeah, I'm inventing Thailand just now, especially around Ko Lanta.

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered

The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

Isn't that simply because axioms that don't lead to consistent conclusions are rejected?

You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.

Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored.

Or in other words: the constraints on maths are imposed from outside of maths.

Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.

Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.

It is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".

> Numbers don't inherently exist

How do we know that it is true?

They are symbols that we assign arbitrary meaning to. They are useful because of the axiomatic framework constructed to support them.

This is an excellent point. When I took algebra as an undergraduate I was blown away by the fact that you can choose any axioms and then derive an algebra based on those axioms. I was blown away because prior to that course I just assumed that our “standard” axioms were immutable.

> choose any axioms and then derive

Sound almost like "jump off the roof and see what happens."

Isn't it the other way around? Axioms are chosen because there are no observable counter examples in the real world.

Not at all, pure math is in part about exploring axiomatic systems that may or may not have a physical counterpart. The latter is immaterial.

Can you give some examples of axioms in pure math that run completely counter to our physical world? For example:

  It is NOT possible to draw a straight line from any point to any other point.
  It is NOT  possible to extend a line segment continuously in both directions.
  etc...

  Things which are equal to the same thing are NOT equal to one another.
  If equals are added to equals, the wholes are NOT equal.
  The whole is LESS than the part.

Note that the original forms of the above axioms "make sense" to us because everything in our physical experience agrees with them. So when you said that the "physical counterpart ... is immaterial", I was curious to see an example of a "physically impossible" axiom.

Most of large cardinal axioms.

> If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered

How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis.

> The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.

Oh no: the axioms come much much later. The order is exactly the reverse one.

Well, I have to agree. From a practical perspective.

The SEP article is an excellent introduction to the philosophy of mathematics:

https://plato.stanford.edu/entries/philosophy-mathematics/

Same question also answered by

Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8

Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4

Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM

I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

Max Tegmark has some fun books. Apparently he got to be at the forefront of some discoveries as he and his wife knew C++ and would do all the after the fact data analysis.

If you've ever been around a campfire with a good storyteller that has their audience at the edge of their seat, I felt like this in some spots.

Entire playlist:

* https://www.closertotruth.com/series/mathematics-invented-or...

Gregory Chaitin was interesting as well:

* https://www.youtube.com/watch?v=1RLdSvQ-OF0

The (apocryphal?) story about Godel not believing in empirical science and only in a priori truths was interesting.

> I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

i came here to give my opinion, but this is pretty much it. i need to read max tegmark's books.

the idea that how we got to the structures and concepts we find is undoubtedly invented, because we've already seen that we've re-invented our methods multiple times. but the structures and concepts don't really change. we extend them, but there seems to be some set of structures and concepts that get settled to. if we found some alien species that has mathematics, we'd probably find that there's some mapping between our structures modulo small choices that were made.

Imagine an interview with a great painter. The painter is asked about the nature of painting and he responds that when he sets out to work in his studio and his brush strokes a canvass, he is discovering the fundamental nature of reality. He doubles down and exclaims that, in fact, reality is actually JUST lines and curves and shades and colors, and his proof is, well, look at how accurate his paintings are! Let's pretend that he actually is a very skilled painter, and many critics have marveled at the extent to which his paintings are indistinguishable from his subject matter. Still though, the interviewer clears his throat after an awkward pause and continues on to the next question.

To me, math is a medium of description much like painting or writing. It's units are not colors or words but points. A point, or "that which has no part", is much finer and carries with it far less baggage than something like a word. Points can be assigned numerical values and played with in clever ways. You can even sprinkle a fine dust of them over anything observable and create a copy of it to arbitrary degrees of precision.

I know math is far more than just the study of points, but I'm not convinced that math is anything more than our capacity to distinguish and describe extended to its limit. I also don't mean to belittle the accomplishments of mathematicians and theoreticians, I just think it's more reasonable to say that math is JUST the limit of description than it is to say that reality is JUST math.

After reading through some of the comments, I think many of us are on the same page.

Agreed

The crux of it is obvious: math is true from the perspective of a human consciousness

It’s defined after cognitive measure.

I won’t say it’s whittled down.

I would say ... it’s sight, sound, touch... attempt at quantifying our qualified experience

All the best math nerds were often deeply gifted and practiced forms of human art: poetry, music, sculpture... and many artists good at math, Brian May, for one example

IMO to get math one needs to build all those abilities

It’s really the people I know that don’t do that see math as a boring toolbox of rules and points

Not saying you’re not that

Math is not a thing like a rock because our ability to behave like more than a rock gave us the ability to define math. It has to be more than a simple this or that

I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering?

We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.

But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.

Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.

In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.

But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.

So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.

> Does it even matter if Math is invented or discovered?

It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.

You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?

I’d say they don’t matter nearly so much as art. The most cosmically significant export of humanity to date is Voyager’s golden record.

Good thing the founders of modern science, mathematics, ethics and political theory didn't share the same opinion.

Ridiculous hubris, to claim to speak on their behalf, to claim to know the opinion of so many and diverse minds, and in such a crude misdirection to boot.

Good thing they wrote volumes about it which have debated and summarized to the hilt.

The summary is that without art, humans are cosmically uninteresting.

I would like some stuff to read on this if possible.

SEP contains dense summaries of every major thinker with extremely high quality references. Random example https://plato.stanford.edu/entries/leibniz/

While art is certainly very important, I believe it doesn't have the same impact upon our lives as math has.

And if you believe art is about creativity, you can find beauty and art in in mathematics and other sciences.

Many great mathematicians were also art lovers, they appreciated music, they appreciated paintings and other forms of art.

I don't think there's a dichotomy between art and sciences.

I agree 100% with everything you say. However, I don’t believe that the “impact on lives” of scientific advances, whilst certainly being of tremendous interest and importance to the humans, carries into much universal significance.

It's the same kind of activity - creation.

The significance of the creation (if it has any at all) is not necessarily the concern of the creator.

Some people do Mathematics for its sheer beauty.

You could say the same about structural engineers.

If you buy into Wittgenstein's rule-following paradox.

You could say anything about anything.

> no course of action could be determined by a rule, because any course of action can be made out to accord with the rule

If I had my own way of (mis?)interpreting him, I think he was alluding to what we now call "strict evaluation" in Programming Language Theory. A diligent rule-follower.

Structural engineering the same kind of activity - creation.

The significance of the creation (if it has any at all) is not necessarily the concern of the creator.

Some people do structural engineering for its sheer beauty.

[Scientific discoveries|Mathematics|Ethics|Political science] are the same kind of activity - creation.

The significance of the creation (if it has any at all) is not necessarily the concern of the creator.

Some people do [Scientific discoveries|Mathematics|Ethics|Political science] for its sheer beauty.

Yes, that's my point. It applies to anything that is creative and dismisses the original point through abstraction.

You didn't make a point. You asked a loaded question.

It backfired on you.

My question asking if you value products of philosophy that have become essential to our society neither contains an unjustified assumption and thus is not a loaded question and still stands and thus has not backfired.

It was loaded.

It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.

It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.

The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.

As a programmer my medium of self-expression is software. I am an artist as much as I am a logician and a scientist.

What I do is create. It also happens to be useful to others, which is why it pays fucking well too.

It literally asks that as a question, posing it as the hypothesis under question.

Your question was answered before you even asked it.

> It matters for people who find it valuable, the same way art, music or literature matters.

Key phrase "the same way".

There's nothing wrong with being wrong, yet you strike me as a person who doesn't like it.

You think that art does not?

> the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?

A major movement (or several, depending on how you slice it) in modern philosophy takes aesthetics as first philosophy - so yes, arguably they matter exactly the same way art does.

you think too poorly of art

I just think our quality of life would be much lower without the forementioned than without art. We'd be much worse off without any of them. My point is not to dismiss art but emphasize the utility of philosophy.

I think not the whole of logic, but this particular distinction -- what changes in the world if one is true versus the other? Is there even a difference that is manifest?

It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?

It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.

Copenhagen requires an extraneous assumption — which may turn out to have practical implications.

At the very least, we know it isn’t parsimonious, though that wasn’t clear until after it was developed (and the philosophical inclination to preserve locality was reasonable).

There is a distinction between logic, which is just the study of formal systems where you can prove theorems as in any other area of mathematics, and the philosophy of maths and particularly of mathematical logic. The same applies for set theory.

The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.

Irrefutable and "unequivically" true if you take some classical FOL as a productive method of producing knowledge, a philosophical assumption. Philosophy came first, mathematical logic is just a formalization of that and the reason it was formalized at all and not somethimg else is epistemological.

Your entire argument attempting dismiss philosophy is philosophy.

I didn't dismiss philosophy.

I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.

Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...

Okay, I read too much into what you were saying. Yes, you're right about all of that.

>It matters for philosophical inquiry.

I was thinking more of tangible implications. Maybe there are some.

Philosophical implications are perhaps the most tangible implications there are: they shape the conception of reality and society that you live in. They only seem intangible because they are subtle and pervasive.

One could replace "Philsophical" in your statement with "Artistic", "Musical" and "Literary" and still have a correct statement.

The construction of your statement implies that you do not think art, music, and literature matter in the same was as philosophy.

> Does it even matter if Math is invented or discovered?

It impacts whether mathematics should be patentable, since patents generally covers inventions, but not discoveries.

Patents cover inventions that are applications of knowledge. For example, chemistry is certainly discovered, but applying the use of a certain molecule as a glue or as a drug is very much patentable. Physics is also discovered, but a special lever system that is a direct application of Newtonian mechanics is patentable. The knowledge that a certain mathematical formula could be written and has certain properties is not patentable, but a device that employs that formula as an algorithm for, say, predicting stock prices, is (in certain countries at least).

I don't see how you are distinguishing between inventions and discoveries. The "discovery" of a mathematical proof is just an application of knowledge, why aren't they patentable? The "invention" of the light bulb, on the other hand, is just the discovery that putting various particles together in a certain configuration produces light in a predictable manner, why is this patentable? I don't see how a clear delineation between the two can be made.

I'm not. I'm saying that devices whose function is derived from mathematics (algorithms) have the same status as devices that derive their function from physics or chemistry. In either case, "application" doesn't mean some intellectual use but commercial use. The purpose of patents is to protect commercial applications in exchange for sharing the knowledge that led to their creation. The knowledge itself is never protected.

When the lightbulb was patented, everyone was free to learn, study, and disseminate the physical knowledge of how a lightbulb works. What you couldn't do is build one and sell it. Whether mathematics is invented or discovered, the knowledge itself is never protected, but if some non-obvious algorithm has a commercial application, it can be patented so to protect building devices that apply that knowledge for commercial use.

But that's the whole point, patents are awarded to things deemed inventions not things deemed discoveries. If you can't delineate between the two you can't say what is and what is not patentable.

Fourier analysis couldn't have been patented, despite numerous commercial applications. It's just the patent system is setup to only reward low-level innovation, so it arbitrarily excludes research level innovation by terming it discovery.

Yes, what is patented is some commercial application of some knowledge, and that application is the invention. It doesn't matter whether the knowledge itself is invented or discovered. When an algorithm is patented, we're no more patenting math than we're patenting Newton's laws when a car brake is invented. We're patenting a commercial application of either, and that must be the invention.

Commercial application is irrelevant to a patent law. Patent applications need only demonstrate "eligibility, utility, novelty, and non-obviousness." Math and other basic research are ruled out on eligibility grounds, not for being non-commercial, but rather for being "abstract ideas."

Also, algorithms are a bad example, as they are just mathematical functions, i.e. "abstract ideas." They should not be patentable. Although I realize patent law is not actually logically consistent.

Commercial application is the motivation behind patent law, and logical consistency is not the point, but rather legal consistency. When you patent an algorithm you do not patent the idea any more than you patent thermodynamics when you patent an engine. You patent a particular application of an idea that performs some function or functions. The algorithm -- or thermodynamics -- stay completely free for anyone to know, study and disseminate. In fact, patents are designed to make the knowledge public so that they could be studied and improved upon.

Whether or not it works is a separate question (and I completely agree that software patents do not perform their role), but patenting a device to predict stock prices is not patenting math, just as patenting a drug is not patenting chemistry.

If it is discovered, then it means there exists math which we don't have to 'invent' and can simply be found like some buried treasure.

I am not sure discovering is easier than inventing.

> But to complicate things a bit, some things don't exist unless we observe them.

Is this intepretation of quantum mechanics still the canonical one?

This is a philosophical question. "No experimental evidence exists that distinguishes among these interpretations."

https://en.wikipedia.org/wiki/Interpretations_of_quantum_mec...

[0] Proposal for an experimental test of the many-worlds interpretation of quantum mechanics

I can't judge whether this proposal makes sense, but there's that.

0: https://arxiv.org/pdf/quant-ph/9510007.pdf

It matters because what if you can't actually describe the underlying foundations of the universe and reality with absolute accuracy as a mathematical equation. We get very close, but if we're always off even by ten to the minus whatever, have we really 'solved' it.

Maybe there is some other conceptual framework that we've not or are not able to cognitively express that underpins things.

to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented

A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy".

Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precision.

Those initial axioms, were they discovered or invented?

Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.

Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.

Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.

and now, in the year 2020, have the axioms been invented or discovered?

>Does it even matter if Math is invented or discovered?

I figure reality is part of math so if it was purely invented we wouldn't exist.

I find mathematics to be the most beautiful thing in the universe, because it is the only thing I can truly think of as being perfect in every way. Existing only as an abstract concept, it still manages to find its way into every single aspect of our lives.

Two completely remote civilizations will still have the same mathematics. Sure one may have a more developed understanding, but if both civs wondered about how to get the hypotenuse of a right-angled triangle, they would both end up with Pythagoras' theorem. The only thing invented in math is our language and representation of such concepts.

Many people believe that mathematics becomes invented the further up you go like pure mathematics and I can understand why this perspective would come through, but take the example of G.H. Hardy who famously said “The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics”. It could be argued at the time that these branches were simply invented mathematics because there was no practical application of it, but 30 years following his death came breakthroughs in cryptography. All of a sudden, this was no longer an invention. Calculus is not seen as an invention but a discovery because of all of its insane number of applications in physics and other areas.

Many areas of math that have yet to find practical use will always come under the scrutiny of being 'invented' but that simply isn't the case.

I firmly believe that all areas of mathematics are practical to the universe and its wonders (and therefore discovered), whether or not we can achieve a level to use such mathematics (or experience it) is a different matter.

I believe that at its core, mathematics is the universe. To say that we have invented it is arrogant and completely strives it of its beauty.

Dr. Hannah Fry's 3-part BBC documentary series, "Magic Numbers: Hannah Fry's Mysterious World of Maths," explores this very question in the first episode, "Numbers As God."

The episode description reads: "Documentary series in which Dr Hannah Fry explores the mystery of maths. Is it invented like a language, or is it discovered and part of the fabric of the universe?"

https://www.bbc.co.uk/programmes/b0bn6wtp

She interviews a number of prominent mathematicians and scientists, such as Brian Greene, and they certainly don't agree one way or the other in the invented/discovered question.

(Alas, I now see that the series is listed as unavailable on the BBC site but I watched it, I think on Amazon Prime or maybe youtube.)

Could the same be said about just about everything?

Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error.

Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discovered.

I guess philosophers must have dwelled on such questions before.

I think I get your point and I‘m open to ideas which drive home your point in a more precise way. However, the main difference between concerning ourselves with whether or not mathematical objects exists vs music or start-up ideas is their place in building a foundation for understanding reality. Science has been a productive program for understanding reality. It happens to be underpinned by math. And as a result, we have encountered some pretty interesting relationships between what the math tells us and what we observe. For instance, neutrinos and black holes were known to exist mathematically before they were ever observed and measured “in reality”. It’s not clear to me that music and start-up ideas hold up to math when it comes to being a tool for understanding or describing our reality, or perception the rig. Math is somehow fundamental to many successful enterprises which seek to describe and explain reality. Therefore, it becomes the point of focus of much philosophical inquiry when seeking to understand reality. Hence the foundational question, do mathematical objects exist? And not, does music or blockchain kittens exist?

>do mathematical objects exist?

Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic.

>the main difference between concerning ourselves with whether or not mathematical objects exists vs music

There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way. For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead).

So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example: https://youtu.be/QzBoGVToWEo?t=342

You can try to answer a simpler question: does the number 3 exist? If you define 3 as "the obvious common feature shared by three cows, three pencils, and three stars in the sky," then the question becomes "does this common feature exist?" and the answer, then, is, well, of course, yes, it does - because it's obviously there!

On a fundamental level music is math.

Well, that's this:

https://en.wikipedia.org/wiki/Harmonic_function

vs. this:

http://openmusictheory.com/harmonicFunctions.html

Maybe, at some fundamental level, they are the same...

Think of it this way: Anyone can invent a new song, and no matter how derivative or lacking in artistic merit, they can still claim to have composed a new piece of music.(La-la-la-laaa. There, I just did it! Short but sweet.)

But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."

There's one thing I cannot wrap my head around.

Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real?

I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life, and even after our minds and consciousnesses form it's hard to believe they are fully autonomous and not simply a function of the processess in our brains we're simply not aware of.

If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.

> Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real?

You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.

What’s depressing about being a nanotchmological robot attempting to conquer the solar system and spread his technology and nanotechnology? It sounds rather exciting and fascinating. Perhaps if the depression is too severe you need to check out your programming ;-)

Mathematics has not been static throughout history. It has evolved at the same pace our science and discoveries have evolved, and as such it has incorporated all necessary elements to help describe new phenomena. To the point that now is able to describe never observed phenomena.

So no, our world is not unreal because it can be described with maths. Maths have evolved to describe it. Same as human language has evolved to support our close environment and our human interactions.

> Math ... describes the world around us to the utmost precision.

It does no such thing. The models we create using mathematical language do.

Math is a precise language that can be used to describe the world around us precisely. It can also be used to describe utter nonsense or utter fantasy precisely.

Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc. Although at least to some extent statistics is the one branch of mathematics that does seem to be applicable (but the way this influences results, as opposed to methods, seems to be mathematically not that interesting).

There have been a lot of efforts of modelling social phenomena, the arts, etc. mathematically and while there are some interesting partial results (e.g. that languages can to some reasonable extent be analysed by parse trees or some aspects of music), most "grander" theories I have seen do not really stand up to much scrutiny.

I'm not an expert, but I could assume that part of this is due to a lot of nonlinearity in the phenomena studies, which means that many classical methods don't work well; maybe chaos theory etc. could shed more light on these things, but I don't know enough about it.

> Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc.

Except it's not ineffective at all in these subjects. Social science experiments have so many variables that the small experiments that can be conducted given the financial resources are insufficient to infer a good model. This is not a math problem, it's a money problem.

If it's ineffective in practice, I consider it to be ineffective.

But even if you could design an experiment perfectly and come up with some strong statistical evidence and then had other means to tease out what is actually causal and what is only correlational, you'd only know what influences what and not necessarily why. But yes, I did say that statistics/probability is some rare exception.

You can study group theory and understand the way physical forces work better, or hilbert spaces to understand quantum mechanics, but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.

> If it's ineffective in practice, I consider it to be ineffective.

Except you have no way to conclude that it's ineffective. Analytical solutions require data to study. Without data, or with little data, what analysis are you going to perform? At best, broad statistical correlations, which is exactly what we find.

You're effectively claiming that spoons are ineffective at a restaurant that provides only forks. Well no, if a spoon were available, then it would probably work just fine.

> but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.

After 5 minutes of Googling:

* Power laws: https://en.wikipedia.org/wiki/Power_law#General_science

* Network theory: https://en.wikipedia.org/wiki/Network_theory

* There's a journal specifically for mathematical social sciences: https://www.journals.elsevier.com/mathematical-social-scienc...

* Economics, game theory, and social choice theory are all examples employing heavy analytical problem solving to social problems

* Quantum mechanics applied to social sciences: https://www.cambridge.org/core/books/quantum-social-science/...

The main stumbling block is that mathematicians are interested in mathematical problems, and so they make a common but mistaken assumption that social sciences either don't have such problems, or they are too messy for elegant math. Take it from a mathematician, this is incorrect: https://www.mathtube.org/sites/default/files/lecture-notes/S...

You may have a point. And I did forget about things like social choice theory or game theory (although I'd assume that partially this is also due to e.g. social choice procedures or "games" often being very limited and artificial settings where by their very nature the relevant space of options/outcomes can be explored in some systematic way, which is generally less the case in more organic, complex settings, such as e.g. gradual societal changes).

When it comes to economics, I know that a lot of people don't agree with the basis of many mathematical models that are used, but I'm not an economist, so I can't speak to that.

So maybe I was overzealous in discounting mathematics for the social sciences altogether. Still, I would contend (albeit with much less evidence):

- There's some measure of people trying to construe "nice models" of things in those subjects instead of trying to make sure they agree with reality. I have a degree in linguistics and I've seen this over and over. Most of these models haven't convinced me at all.

- The amount of maths that you either need or at least benefit from in order to do good research in such areas is still substantially lower than in, say, physics. I think it's still to be noted how we can describe much of physics just with a small number of economic and elegant models. I haven't seen anything comparable in any of the social sciences.

https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

> If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.

It can also be liberating and uplifting. You're a part of the universe - you are the universe expressing itself. The result of uncountable generations of brutal evolution, you're quite a miracle!

>I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life

Free will doesn't depend on the universe being deterministic or not. Things that haven't happened yet are likely going to happen in a certain way based on the trajectory, but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.

>but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.

This is of course debatable, there is some evidence to suggest that the feeling that you took a certain path is illusionary. Your brain made the decision, then you became cognizant of the options and you felt yourself making the decision[0], but if you could rewind the universe, you would always choose the same path based on your 'brain-state' at that time. At least that is what I understand of the most extreme "free-will doesn't exist" position.

Just got lost in the SEP rabbithole [1], still not sure where I land on this issue.

[0] : like how you feel thoughts 'bubble up' during meditation, you didn't actively 'think' those, they appeared to you. EDIT: atleast that is what it feels like

[1]: https://plato.stanford.edu/entries/freewill/#DoWeHaveFreeWil...

That things would repeat after a rewind, given the same "universal state" just contradicts free will from the perspective of looking backwards (to the then present), not in the current present or looking into the uncertain future.

That's exactly the idea behind Max Tegmark's theory which he described in his book: Our Mathematical Universe

https://en.wikipedia.org/wiki/Our_Mathematical_Universe

Math models are real things, hence reality also.

Recently I watched Sixty Symbols video on why light is slower in glass and there are multiple explanations on it.

What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk

On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality.

That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable.

Wouldn't that mean that mathematics is just an invented tool to reason about physical models?

Well yes and no. A lot of mathematics found its roots in trying to model the real world, however I would then argue that the truths we prove about these models are truly discoveries. And with these discoveries we can often generalize the setting to more abstract formulations, independent of the physical reality.

I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical.

This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.

I'll never understand how someone otherwise so apparently intelligent can be so religious. Weird, the systems that we spent massive amounts of energy designing to precisely describe reality do that better than all the ones that we threw out along the way!

> I'll never understand how someone otherwise so apparently intelligent can be so religious.

Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance.

I'll never understand why some people believe science and creationism can't go hand in hand.

If you want to understand why some people believe that science and creationism don't go hand in hand, then studying epistemology would help you do that. Most introductory texts will cover what you need.

Sounds like I got something new to learn during this lockdown, thank you. After a quick look at the wikipedia page on epistemology (very interesting), I'm not sure how this explains why some people believe that science and creationism don't go hand in hand.

A super super short version is about the question "how do you know X?" Its generally (although not universally) agreed that there are two types of knowledge, a priori and a posteriori. Roughly, in the first case, all you can deduce is implications from assumptions. For example, "if X and Y are true then Z is true." Such things don't require experience. The second case requires evidence gained from experience. Mathematical knowledge is of the first type, scientific knowledge is of the second type. The school of rationalism prioritizes the first type and the school of empericism emphasizes the second. That the universe was created by an omnipotent omniscient and benevolent god is not an emperical fact but requires some kind of a priori supports. Thats why you see so called proofs of god rather than scientific theories of god. There are other options of couree like mysticism, the claim that knowledge can be gained outside of these methods, or an emphasis on faith, that we don't take these things to be justifiable.

I probably butchered this but it gives you some starting points to research and hopefully shows a model of the debate. Huge topic. Check out https://plato.stanford.edu/entries/epistemology/ as a hardcore crashcourse or check "crash course philosophy" on youtube for the relavent sections at a more HS level. SEP has entries on god, faith and mysticism as well as rationalism and empericism too.

Careful with your use of the word creationism. I guess you mean it in the sense of the very vague and general claim that “god created the universe”, but in my experience it is used to refer to a much more specific set of claims, such as “the Earth is about 6000 years old”, which is absolutely not compatible with science.

According to Wikipedia: 'Creationism is the religious belief that nature, and aspects such as the universe, Earth, life, and humans, originated with supernatural acts of divine creation.'.

I choose my words carefully. I get what you're saying though, but I think that's more of a subjective matter.

And who/what created God -- e.g. who is God's God? And so on? It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.

Why is there a need to insult anyone when it is you yourself that lacks understanding of the matter. We believers believe that God was uncreated, He is unlike His creation. Only created things have a creator. And God created time itself, so there is no before or after Him. He was always there. So there's no other answer than "that's impossible" to your question.

Of course this is hard to understand, but what would you expect? We can't even comprehend the tiny amount of His wisdom He has given us, let alone comprehend all of it.

If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there? Yet you believe that living things, which are not even comparable in complexity to an iPhone, were formed by a chain of coincidental events in the universe?

You can see God if you are willing to, unfortunately most atheists keep their hearts closed...

You ascribe beliefs to me via stereotype. I don't "believe" in the Big Bang Theory. I'm content knowing that I don't know. It doesn't strike me as rational to speak in such absolutes about subjects which are beyond human comprehension. To each their own.

> It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.

> You ascribe beliefs to me via stereotype.

Not that I feel offended by the first quote, since I don't believe in a hypothetical god, but how am I the one stereotyping you? If you don't believe in God, which you made clear in your first comment, then you disbelieve in Him. So that means you're an atheist, right?

Or am I misunderstanding the concept of being an atheist? To each their own of course.

> If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there?

The evidence says that actually did happen. Abiogenesis and the evolution of human beings were just part of the process of producing an iPhone through a "mere accident" of natural forces over the course of the last 4.5 billion years or so. Producing an iPhone was never the goal, of course–natural processes don't have goals. But it was one of the consequences.

If God can exist without being created, so can what you refer to as "His creation". This idea that the visible universe requires a Creator, who in turn does not require a Creator, is arbitrary and capricious.

Last week I came across another fascinating podcast by Penrose that you folks might also like: https://www.youtube.com/watch?v=orMtwOz6Db0

Thanks for the link. At 1:02:57, Penrose proposes "infinite cycles of the big bang"

https://youtu.be/orMtwOz6Db0?t=3777

Is that multiverse theory disproven by Wolfram's latest blog post?

"could there be other universes? The answer in our setup is basically no."

https://writings.stephenwolfram.com/2020/04/finally-we-may-h...

If B follows from A via logic, but A is invented, isn't B invented as well then?

Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered.

However to me the basic premise, take some objects which aren't numbers but which square to a number, feels very much like an invention. So as such wouldn't the nice results be inventions as well?

[1]: https://bivector.net/doc.html

> If B follows from A via logic, but A is invented, isn't B invented as well then?

Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.

Yet if I pull the trigger we say that I killed someone, not that the bullet did.

People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.

It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.

Well if one would consider numbers and their product "discovered", then my point was that introducing the objects which have the property that they are not numbers but square to numbers seems to me like an invention.

At least if there is to be any meaningful distinction between a discovery and an invention.

Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.

You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results.

The world of ZFC doesn't feel very constrained by the actual universe, is all.

And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.

Sure, math/ZFC can describe potential and impossible universes including ours. But the motiviation for its invention as a language was very likely to understand relationships in our reality, physics, trade etc.

A vaguely, but tangentially related fact, is that Stephen King actually considers the stories in his books as discovered, rather than invented. He talks about it in his book 'On Writing'. He considers the stories to be sort of preexisting things, and his job as a writer to unearth and discover them, rather than invent them. It's about his frame of mind while writing, I guess, but still.

It's a common theme among sculptors too: the shape is already in the stone, they just have to uncover it.

Is it? The only time I have ever heard this expressed is as a bad joke.

I know from a few general humanities classes I took that it was a common way of thinking about sculpting during the renaissance. And modern learning material on the topic presents it as a visualization aid. Let me be clear: It wasn't some mystical way of thinking there was, literally something that existed in the material. Again, it was more of an aid to thinking about the endeavor.

I doubt that he actually said that but I could be wrong.

This is actually a big plot point in The Dark Tower books.

He describes stories as 'relics'. Here's a direct quote from the book: “Stories are relics, part of an undiscovered pre-existing world. The writer’s job is to use the tools in his or her toolbox to get as much of each one out of the ground intact as possible.”

The book is pretty good, well worth the read if you're only slightly interested in the topic!

To me this sounds like an echo of the view that free will is an illusion. The stories that King is going to write are already predetermined by the laws of physics applied to the initial conditions of the system. In a sense, he's really just "along for the ride".

We're all slaves to ka.

What can be imagined is what can be discovered.

The etymology of invent has the terms 'contrived' and 'discover' baked in. if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover, we find that it is rooted in the ability to 'compare' and 'imagine'. From this we can then formulate the opening statement.

My take on this is that Mathematics is such a vast area of investigation - much larger than Nature itself - that there is in fact a mix of both. In that regard it is close to Engineering, where in an attempt to design something on might stumble upon things or perform some types of research and discovery, and vice versa.

It is a dispute in ontology. Do mathematical objects (say, numbers, sets) exist in the world? Some say, yes; others say, no; some others say, they exist in another world--called Platonic world.

We see similar disputes in philosophy of (natural) sciences: for instance, instrumentalism doesn't subscribe to the ontology of realism.

I read this book about it: https://www.amazon.com/God-Mathematician-Mario-Livio/dp/0743.... Won't spoil the answer, but was nice to go through the evolution of Mathematics.

Put me into the 'discovered' camp.

In most fields, you have to invent tools that enable you to discover things. My guess is that this is also true in mathematics. We get confused sometimes because the tools often happen to look like the discoveries because of all the fancy greek symbols.

The answer is: "both".

Where do Mathematicians look to discover Mathematics? In the depths of their own minds.

The part where you "look and think deeply" is discovery. The part where you "express your discovery in a coherent language" is invention.

I initially read the title as "Is Mathematics haunted or discovered".

Operational theory of mathematics: you define (invent) some rules, of starting points and ways to change them. Like the rules of a game.

In a sense, all possible outcomes already exist (only to be discovered).

Phenomena are discovered. Mathematics are invented to describe them.

Not all mathematics though.

Math works by starting with a set of axioms. These are chosen, usually but not necessarily such that they will describe some kind of physical reality. That is the part which you invent to describe phenomena, like the Peano axioms which describe quantities of countable things.

But then, you discover (and of course also prove) things that follow from these axioms. So a mathematician would find that if you take this set of axioms, then this statement is true. Is that an invention or discovery? I would call it a discovery, although different from a discovery of a natural phenomenon.

Geometry and arithmetic and calculus all existed before axioms and worked okay. You can do quite a lot of mathematics "naively". It's only in the 20th century that axiomatization became really important, because naive set theory turned out not to work very well.

And what if all phenomena are intrinsically mathematical? Then mathematics is discovered as well.

Like in any game, you invent the rules (axioms and logical systems) then you discover the consequences (theorems), just like you invent a maze then you discover the way out.

The platonic universe of the entirety of math sounds (this week at least) like Wolfram’s “Universe of Computational reality” of which the physical reality is just a subset.

Invented to describe what's being discovered.

You have to add: discovered inside itself.

The linguistic conundrum here arises due to the recently adopted mindless use of the word 'invented' instead of 'created' or 'designed.' (Indeed, Linus created Linux rather than 'invented' it as many would say today.) Invention is a form of discovery. So, the question should be, "is mathematics created or discovered?"

This is a silly semantics game and a boring one at that. the presumption that the words are contradictory, or that mathematics is a monolithic entity that plays a single role of invention or discovery, is a poor way of exploring the social and cultural (co)evolution of mathematics and methodical experimentation and observation.

You can't just say this to people who keep asking this question, and have been for ages.

Why not? It’s not like they have any obligation to respond, and someone needs to point out the bullshit smell wafting through the room.

A bit both:

e.g. The circle is an invention. But the number pi (it's value) is discovered

What do you mean when you say that the circle is an invention rather than a discovery? Are you drawing a distinction between the mathematical ideal and the arbitrarily close approximations which can be found in nature? Though by that criteria that case pi must also be an invention…

The expression is invented but the underlying relationships are discovered.

It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter.

It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.

It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.

They are necessary for the continuum.

If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.

It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.

Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.

Complex numbers have the same cardinality as the reals.

That's true, but irrelevant.

Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.

I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.

> If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.

Then again, have you ever observed a non-computable number as a component of the value of a wave function in nature? Real numbers add a lot of cruft which can never be practically observed (due to not being computable).

One of the main reasons we lose things is that our results have been built upon the real numbers since they were easier to conceptualize and to work with, but it's possible that a lot can be recovered (maybe even everything we care about) using only the computable numbers. For instance, see https://en.wikipedia.org/wiki/Computable_analysis for some results in recovering analysis (limits, differentiation, integration, etc) using the computable numbers.

Don't they, though? The path of an orbit in vacuum should follow pi, unless coordinates are discrete (which they might be, I suppose)

You could argue that it is only the model we construct using maths that contains pi.

The real question would be whether all simple, precise models of the concept contain something that can be identified as pi.

Pi can be a unit of one in an alternate numbering system.

Many reals are constructible so they are accessible in the sense that it's a theory of phenomena I can act and build upon, and the theory discusses up to an infinite number of cases even if I won't build out that far.

Depending on how philosophical you want to get, it is not so clear what exists and what does not. Neither it is absolutely clear what means to be discovered.

I think that ideas can be discovered. In my opinion, theorems, or more precisely the proof of these theorems, for instance, are discovered. In the sense that they are not an invention, they "emerge" from more fundamental definitions.

You seem to be of the opinion that the only things that can be discovered are those that emerge from the "real world", whatever the real world is. That is, I guess, an acceptable interpretation, but I do not think is the only possibility.

To give a very concrete example, chess rules do not exist in the real world, but knight's tours are discovered.

On the contrary. I think that "real" and "imaginary" numbers are both discovered and neither concept has more reality than the other.

I think that even chess rules have a meaningful "existence" (in a Platonic sense) even if they are in some sense arbitrary. We can't discover "the one true rule system" but we can discover various possible systems and also theorems about them.

Why aren't real and imaginary numbers both invented?

If aliens ever draw a square and wonder what it's diagonal is, they'll have to face the idea of irrational numbers.

I read that we can advance physics the most by research into mathematics. Sooner or later the oddest math finds application in nature, which is odd. See also: https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

Negative numbers do of cause have their application in nature (negative charge etc.). But it is possible to do correct mathematics that have no interpretation in real live. E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.

> E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.

The only trouble here is in the assigned interpretation. You do have -4 students left compared to the initial state but there is no reason to assume the initial state was 0.

The personal expression can get very far too, further than perhaps an advanced alien race may know. Supposing an alien race never discovered category theory, their mathematics could still be more advanced than ours, but perhaps more verbose, longer to write down or otherwise bulky and inelegant. The question of whether elegant mathematics is more advanced I guess becomes tricky to answer. But pedagogically and for practical application, elegance is paramount.

I think the most celebrated mathematicians are such because of their personal expression. And that "style element" also inspires people to go further than before. Grothendieck, who featured here a few weeks ago, is a case in point.

Sad thing is it seems likely that, at least from some vantage point in the universe, we're the aliens that discovered category theory.

> It's silly to think that the sqaure root of negative one is something real to be discovered.

As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".

Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?

1. Isn't an infinite universe consistent with observations?

2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.

3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.

4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.

> The same for negative numbers, for that matter.

It might be expressed differently, but the concept of addition seems very fundemental, and additive inverses (negative numbers) are a very real part of addition.

> It's silly to think that the sqaure root of negative one is something real to be discovered

If we assume the aliens also have a need for multiplication, combining it with multiplication we get polynomials, and to factor polynomials you need complex numbers.

Besides, if they think of quantum mechanics they'll surely need some way to express what we call the complex numbers too.

Even if they don't think of them as "numbers", they'll absolutely discover analogous concepts and theorems.

"additive inverses (negative numbers) are a very real part of addition"

Inverses are just an abstraction that helps you rearrange operations.

You'd think something like accounting would need negative numbers, but nope, it gets along just fine without them.

They have the concept of "minus", which is just another way of adding with a negative.

Also deficits and surpluses seem like normal accounting terms.

"They have the concept of "minus", which is just another way of adding with a negative."

You have chosen to generalize it into the anstract concept of negative numbers. That doesn't mean that negative numbers are a fundamental truth about nature to be discovered.

Think about it like functional programming. Is that discovering some natural law of computing? Or is it inventing a new way to express computation? Of course it must be the latter, because you can compute anything without the use of functions.

>The same for negative numbers

Are you sure negative numbers are not "real" in the same way natural numbers are?

That would mean particles like electrons with a negative charge are not "real", too. And someone can argue that his negative bank balance is not "real".

As for complex numbers, I fail to see how they are less real than real numbers. Real numbers are points on the real axis, while complex numbers are points in R^2 plane. If we question the reality of real numbers, we can also question the reality of the plane.

Mr Euclid would frown to hear that.

While negatively charged particles are real, the 'negative' charge is a way to distinguish them from a 'positive' charge. It's a naming convention more than it is an actual negative value.

We could easily call them 'black' and 'white' particles and the system itself would stay the same

This is not true. You're losing a crucial property in your translation, namely that the negative numbers are the additive inverse of the positive numbers. In other words, it's essential for electrodynamics that a + (-a) = 0. You could add this requirement to you 'black' and 'white' terminology, but then you've just arrived at negative numbers under a different name.

Don't let the naming confuse you. The name "negative" might be a peculiar, human-specific thing, but the concepts of addition and additive inverses are not.

This is just the age old question of nominalism vs realism

Of course..we invented the ‘language of mathematics’.

I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind Audiobook.

Java was invented, Agda was discovered.

(Obviously I'm in the religion where math does exist.)

Thank you for sharing this, this an amazing interview series !

Fallacy of equivocation.

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