Intuitionism
en.wikipedia.org
en.wikipedia.org
I often entertain the idea that all the patterns we observe are merely things that match our capability of understanding. This could explain the "unreasonable effectiveness of mathematics in the natural sciences".
It may also help to guide us away from the "why is there something rather than nothing" problem. If existence is total chaos, then we as humans could be limited to the hyperplane of our own observable patterns, which fools us into thinking there is some inherent order -- which there isn't. This leaves us with "why is there chaos rather than nothing", so I doubt it's of any help :)
Great ideas to ponder, but rather hard to reason about.
(Edit: To avoid thinking that I'm a crackpot, with "capability of understanding", I am referring to the physical processes that lead to the existence and dynamics of neurons, not to the platonic world of ideas on top of that. If someone could point out how unoriginal or nonsensical this idea is, it would save me from writing a blog post about it.)
I have a similar but somewhat different take on this. The universe comes first, it behaves - for what ever reasons - in the way it does. Than we humans show up and invent logic, a set of rules that is useful to reason about our universe. It either rains or it doesn't makes sense in our universe. But the universe could potentially have been different, for example something like the many-worlds interpretation but where the inhabitants of that universe can experience all the branches. Their logic might say it rains and it doesn't.
Other ideas like objects, properties, space, time, causation or countability might also be influenced by the way our universe works and how we perceive it and they might be far from universal or useful across different possible universes. On top of that we than construct mathematics and explore what all the things are that we can build from those ideas and our laws of logic. It should probably not be especially surprising that we can on the one hand construct things that are not realized in our universe and on the other hand find structures that are useful for describing and understanding our universe.
We could reduce the requirements down to an implementation of a Turing machine, or something similar. (For the argument I simply ignore whether the machine is conscious -- that seems irrelevant in this context.)
That still requires some kind of discrete switch, which may seem fairly minimal to a human observer, but in reality consists of tens of thousands of atoms to operate. Atoms used to be simple, but turn out to be quite complex as well.
Representing, say, a circle in a fairly minimal system such as this would probably require the cooperation of millions of atoms.
Bold claim :)
Even while maintaining a willful agnosticism about Platonic realism, it seems clear that the business of doing mathematics - intuitionistic or otherwise - depends on the ability to state and follow unambiguous rules, or else how to establish a proof within some axiomatic system or other?
But if mathematicians have this ability, even as an asymptotic ideal, and are able to make judgments as to when rules are not being followed correctly, mustn't this in turn depend on some pre-existing regularity (i.e. order) in the universe? Ineluctable physical law seems a natural candidate for this order.
I noticed that in nature (i.e. in the physical universe) there appears to be no logic. There are not even two things exactly the same, as far as I can tell. So that lead me to believe that "counting" (or abstraction) is not even a property of the universe, but possibly only a human (or animal, or Turing machine) construct. To me, logic only starts to occur when a very complex amalgam of matter comes together [1] to realize a discrete switch. Using these discrete switches, organisms build memory and abstraction mechanisms, start counting, and do mathematics.
So my thesis is that things that are considered to be "basic" by most, such as logic or mathematics, are in fact quite specific systems built on top of chaos. From there, it remains to be proven that all physical laws that we observe are no more than projections of the chaos onto such a system.
Perhaps a metaphor that helps to take on my perspective, is to look at a screen filled with random noise, and then observe some patterns in there. Now replace the screen with an infinite dimensional set of chaos, and then observe a pattern that is our universe. With the added twist that we are part of this chaos, and observing the pattern, possibly in the form of physical laws.
Of course, there are many problems with this theory. What does the chaos reside in? How can there be discernible parts in the chaos? Is time an emergent property inside the chaos? How can it be that the patterns that we observe are so consistent?
However, to me this theory seems more fruitful than merely accepting that we cannot say anything about things that science cannot observe, or that some deity created all this.
[1] With "comes together" I do not refer to a dynamic process, but to the accidental occurrence of stuff in such a shape or form. Obviously, my ridiculous theory asserts a chaos chock full of dimensions, where time and space are but supporting actors.
Aren’t all basic particles defined by the fact that they are exactly the same? And countable things rely on some difference, such as different spatial locations — or else they wouldn’t be countable, they’d just be the same.
While two neutron stars are distinguishable they are also classifiable as a real type of star in a manner that seems to go beyond human perceptual idiosyncrasy.
But I'm a deep Platonist/Pythagorean — so my bias is that "all is number" and the world is made of math. Math is real :)
In general, your view seems like a definitional issue to me. If you want to call what the world is made of “math”, then what you and I mean by math are two different things, and using the same word to describe them only leads to confusion.
I can make up all sorts of math that isn't really "discovered", it's more "invented". Most of it won't correspond usefully to the physical universe. See e.g.: https://plato.stanford.edu/entries/formalism-mathematics/ :
> "mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."
I take the brevity and lack of commitment to a position in your reply as an indication that you're not really willing or able to defend your position. That's fine, it's a complex topic, as the many articles on the SEP linked above attest. But if you want to claim "the world is made of math", then the onus is on you to define what you mean by that. To me, it looks a little incoherent.
Plato described math as a realm distinct from both the physical world and the world of consciousness. This doesn’t support the idea the “the world is made of math”.
Newton described the world as operating in accordance with the rules of math, but that’s not the same as being “made of math.” Plato’s view is compatible with this: what Frege described as the “third realm”, the realm of abstract objects, can have a relationship with the physical realm without requiring that the latter be “made of” the former.
Aristotle explicitly distinguished between physics and mathematics, saying in his Metaphysics that physics is concerned with things that change, whereas mathematics encompasses things that are eternal, do not change, and are not substances. So Aristotle seems to explicitly reject your view.
As such, I don’t accept your claim that your position is “one of the oldest and most influential ideas of all time.”
> But the onus doesn’t lie with them (or me).
If you make a claim, the onus certainly lies on you to support that claim.
Newton was a Pythagorean. He even attributed the inverse square law to Pythagoras.
Plato is always hard to pin down, but he describes the immaterial world as crafted by number, prior to the material world.
I don't think you've taken on the full force of the argument that regularity in human activity (such as building systems) requires a source of order for it not to simply dissolve into chaos itself.
> How can it be that the patterns that we observe are so consistent?
How can we claim to discern consistency (or inconsistency) without the ability to follow a rule correctly? And how can we follow a rule without a source of order or regularity in the cosmos? Wouldn't it be like trying to build a the Eiffel tower out of live slugs?
If you insist on an absence of order in the physical universe, the onus is on you to explain how regularity in human activity (required for mathematics of any kind) can be achieved without it.
This is not an argument for Platonic realism, BTW, or against intuitionism, roughly construed as the view that "mathematics is a creation of the mind" as per [1], or in your formulation that 'mathematical abstractions are not part of the physical universe' (if I understand what you're saying). You can perfectly well believe that mathematics is a mental construct and at the same time acknowledge that it's possible to observe regularities and order in the cosmos. If you want to insist that the regularity doesn't come from physical law, then I find it hard to see how you'll escape from some kind of Platonic belief in a non-physical realm that serves as the source of order :)
In your TV screen dots analogy, isn't it usually thought that patterns appear only because of the structured, generative activity of law-observing physical components, specifically the neurons comprising your grey matter?
Entering purely theoretical space here: On the timescale of eternity this might be a local pocket of some logical organization but there is no fundamental logic governing everything. Our observation is limited so we can’t perceive chaos, instead evolved to only recognize patterns. Over infinity, pure chaos does not preclude long pockets of what looks like order. What we consider fundamental rules could very well be local phenomenon, which we are a product of.
Of course this purely theoretical - all I’m saying is that intuitism could be true while also math being useful to predict things right now. We could also only exist for an instant and all our memories just construct, but that’s not very useful. It’s more useful to believe in scientific method because what’s repeatable is provable, whereas chaos is by its nature unprovable - which doesn’t make it impossible.
It is useful to focus on repeating things, and useless to focus on randomness. But I don't think it's necessarily true that randomness (probably a better word than chaotic since chaotic systems are complex mathematical interactions) doesn't reign. We evolved to take advantage of repeatable things, our sense organs and perception are all focused on things that are repeatable. Our definition of usefulness (what is useful/what isn't) depends on repeatable things. I believe there's a pretty high likelihood that we are blind to anything outside of that, such as true randomness. IE, we literally cannot conceive of true randomness since we are products of an environment that rewarded it.
To your point, I guess this isn't exactly Intuitionism, since Intuitionism says it's a totally human construct, and mathematics has provenly predicted things in nature from purely theoretical models, whereas I just find the part that supposes mathematics isn't a fundamental part of objective reality possible.
Either way, I don't think it changes anything about how we do math or science or anything - how could you even study this? By definition understanding and using things depends on repeatability. If there truly were cracks in it, they by definition couldn't be repeatable. It certainly won't help us get food.
EDIT > If you don't consider it meaningful, then why are you having it?
I just find it interesting since I've had this thought before and seeing what other people think of it.
For reference, my views are somewhat related to "emergentism", "connectionism", and "realism", but I haven't found a school of philosophy that I feel comfortable with.
> how could you even study this?
This is indeed the biggest challenge. I am currently studying this from a conceptual art perspective, because philosophy and science do not seem adequately equipped for this kind of problem.
With respect to the full force of the argument: I assume that the "regularity" stems from the physical systems that make up our brain. Just as replication through DNA offers some stability in life, the shape of our neurons (and perhaps the dynamics of space-time, and the laws of physics) offer some regularity in the chaotic universe.
In my view, this regularity is but an accidental blip in the totality of existence, but to us, who cannot observe the rest of the chaos, it seems fundamental -- which from the universal context, it isn't.
The biggest problem that I cannot get around is that I somehow assume this chaos exists, and allows for things to exist inside of it. I do not know how to provide arguments for that, other than the negative one that it seems highly unlikely that "there is something rather than nothing". Likeliness, and the fact that I can define these abstract concepts, only make sense in the realm of human thought, so I am sort of stuck in a recursive loop there.
With regards to the second part of your reply, again it is us humans who do the discerning. It is an emergent property of our brain (or possibly of slightly simpler, but still rather complex "discrete switches") that we can discern things. In the underlying universe of total chaos, there is no context, no logic, no measure to discern things.
So, the source of order does arise through physical constructs, that happen to have a certain structure that allows observation. It is humans, mammals, octopuses, computers, that can use this universal form of observation to process input, and then do observation as we know it. So I suppose my idea is some kind of realism, but my reality is nothing more than pure and utter, unbounded chaos. And we live in some corner of that.
The grand claim is that mathematics is nothing more than the result of some self-observing shapes in the chaos that is existence.
Again, I feel sorry for all the readers who try to make sense of all my overloaded concepts. I wish I had the skills to write down my thoughts more rigorously. Or perhaps someone can save me a lot of time [1].
https://medium.com/physics-as-a-foreign-language/how-do-we-k...
https://www.popularmechanics.com/science/news/a27731/what-if...
The first link is an article that explains that all electrons are exactly identical, suggesting that there are indeed things in the universe that are exactly the same. However, the second link discusses the one-electron idea by Wheeler, which suggests the exact opposite :)
That is a common definition of identity, two things are the same if all their properties are the same. So by definition there can not be two different things with all their properties the same as this would make them indistinguishable and therefore the same thing. But if you relax this a bit, then for example elementary particles like electrons are - as far as we know - all completely identical up to their position, momentum and spin.
Cogito sicut tu cogitas, ergo sum sicut structus es: I think as you think (at least for a moment); therefore, I am, at some level, structured as you are (at least for a moment); and therefore, there is order in the universe, at least in the temporary alignment of our structures that enables us to think along these same lines?
(Saying nothing, of course, of whether you "exist." I may be a brain in a vat, and your thoughts may be the emission of an evil demon running a simulation — but that still means that my thoughts and the evil demon's simulation share structure that implies an underlying shared set of computational axioms!)
Yes. And such order exists within the observer, the mediators. We are measuring instruments. Scales are bound to the ruler and not to what’s being ruled. So mathematics represents order only in as much as there are people to validate it. Should all rulers be broken and forgotten, then there’s no measure at all. This is the same for those orders not yet established, that is, the future of science. Nevertheless, I do believe there’s a principle which allows order, but it’s not order itself, but a foundation of order, which I believe to be Unit. Unit is not mere duality, because duality implies two, Unit would be more like Cause-Effect, wherein one is the same as the other (either Cause-Cause, or Effect-Effect, doesn’t really matter). This is also different than yin-yang, since each is discrete, and discreetness itself cannot exist prior to Unit. I like to think that all numbers are qualities of Unit, and the whole of mathematical theories are different theories of Unit, so they will be consistent every time Unit is maintained consistently, when something “follows” from what has already “followed”, following some definition of “following”, whatever it may be. It would explain the effectiveness of mathematics in the sense that the whole Universe is Unit out of self-similar Unit. The fact that all of information can be encoded in 0s and 1s is a great illustration of the power of Unit, and the fact that binary streams only makes “sense” upon interpretation is a great illustration of consciousness, which is an expression of Bias. Even if the Universe would change so all of the physical “laws” would mutate, Unit would still persist unchanged. New things would still “follow”. I haven’t come across anyone realizing that a theory of “everything” can exist but be useless, just like the concept of “everything” is useless as a particularity, and theories aim to be particularly applicable, so a theory which really applies to every thing applies to no thing. This would be the utmost conclusion of Gödel‘s incompleteness. How would a theory of absolutely everything be different than an infinitely long ruler without any subdivisions, or even with infinitely many zero-spaced subdivisions? One wouldn’t be able to measure any particular thing with such universal ruler.
I think I lost your train of thought when you say that numbers are qualities of Unit. Does your universe involve only "Unit", or are there other principles at play as well?
[1] https://www.reddit.com/r/holofractal/comments/6wnekw/this_be...
Consider the set of all things, including the ordered and disordered. Consider the operation of taking subsets of that set. Consider that conscious entities such as us only arise in ordered subsets (for some definition of ordered).
Those conscious entities would see their proximal environment as ordered. They might assume that the only things in the set of all things are those things that are ordered in the same way as the proximal environment from which they arose.
Our universe may be just a subset of a subset. That is, our universe might be one kind of universe that has some apparent quantum-field or relativity-geometry based regularity. There might be some number of universes that share that kind of ordering and perhaps not in all of them conscious entities arise.
It is still an interesting question to ask: why would conscious entities arise within this particular kind of ordered universe? But it no longer remains the question of whether or not regularity is a fundamental property of all possible states of existence. And in fact it leaves open the question as to whether or not some kind of consciousness (perhaps very different from our own) could arise in what would appear to us as chaotic universes.
If our world would be an "accidental" subset of the universe, with ordering, then how can it be that this ordering is so consistent? Would it not be more likely that we'd live in a world that has only some order, or varying order depending on one's position in space and time? For example, I would expect a lot more miracles to happen, but every physics experiment turns out to be extremely consistent.
This led me to believe that there is another process at play. My thought experiment now assumes that observation enforces a certain kind of consistency in the laws of physics. That is, the systems that we use to observe something, must by their very existence result in very consistent patterns in the chaos. What this precisely looks like and how it operates is left as an exercise to the reader.
That sounds like a typical chicken-and-egg problem.
> how can it be that this ordering is so consistent?
One thing that I didn't explicitly state is that the universe only needs to seem ordered. You are making an assumption that isn't necessarily founded. 99.9% of the time we aren't checking on the ordered state of the universe. It is the exceedingly rare case that we measure with sufficient granularity to expect to see quantum effect, for example.
> I would expect a lot more miracles to happen
I totally understand this. You might expect a table upon which your keyboard rests to change color as we pass through a chaotic portion of some multi-verse. But that assumes order can't exist at the time-scales of universes.
Imagine it like a picture. The set of all possible pictures at 640x480 is massive but finite. The vast majority of images in that space are like white noise. But you've probably seen millions of images at that resolution that are totally coherent. You don't expect the middle block of some specific image to randomly be a different color. Same with a movie. The set of all sequences of 640x480 images at 30 fps with 1 minute duration is finite. It is full of total chaos. But when you watch a movie you don't expect chaos to ensue at any moment.
If you consider our entire universe like a single image from the set of all images or a movie from the set of all movies, it isn't surprising that it is entirely ordered from beginning to end. It just feels weird because the scale in both time and space is so large in the case of the universe. But given the scale of the entire set, such a large subset having internal consistency isn't totally unreasonable.
What Plato called forms are just descriptions. We have a description of what a circle is, and anything that matches that description is a circle.
You haven't really come to grips with what a world that was mostly inconsistent would look like.
Probably one that had trouble evolving high-order lifeforms, aka anthropic principle.
Thats because chemicals shape/influence our personalities and emotions. You'll see the same consistent behaviours in animals as you do in humans, when given the same chemicals.
https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RS01....
Your mobile phone can be used as a weapon against you, triggering calcium waves in the body which could be harmful to your liberty and your life.
I know because I've had it done to me!
(Yes I know that one cannot use "reasonable" as an argument for a context in which human arguments do not apply. The entire thing is a thought experiment, nothing more.)
Still, I've come around basically to mathematical platonism. The structure is out there, we just happen to be smart enough to tease some of it out.
I have two arguments for this. The first is the very existence of long-standing problems, and their eventual resolution either way. For centuries we were able to wonder whether Fermat's last "theorem" (which was really a conjecture at the time) was actually true, and eventually Wiles came around with an extremely complicated proof and it was settled. And we do believe that math/logic is consistent enough that someone else couldn't just have followed a different train of thought and come up with a proof of the opposite. How does a strict intuitionist account for this kind of situation?
The second, and possibly deeper argument, has to do with structural equivalences. I've been out of the field for decades, but I know that a standard trick in academic math is to develop structural equivalences between disparate fields. You want to prove something in an area of math, but it's hard, so you prove that the whole structure of that subfield has a one-to-one correspondence with the structure of another subfield, and then prove the corresponding theorem in the other subfield, which happens to be easier (see: analytic number theory). Again, this sounds like exploring an existing territory, not like arbitrarily building thought-bridges here and there. The bridges are where they are, and if you try to build one where reality didn't put it, your proof will get nowhere.
An even stronger form of this is that, in advanced mathematics, all kinds of notions of universality appear all the time. One of the most famous is probably computability theory. Just using a few basic symbols (say, integers, first order logic and some additive operations), you get theories of varying power. But as soon as you hit a certain level of richness, bang, all of a sudden, you've hit computability. Your theory is rich enough to embed a Turing machine, and therefore is exactly as rich and expressive as any other computable - even if one is based on numbers and multiplication, and the other on graphs or some such other weird thing.
Universality shows up in lots of places. I'm too far out of the field to remember them, but it starts with the very integer numbers - there are plenty of ways to formalize their initial construction, but the eventual result is exactly the same.
At this point my general thinking is that the bulk of the structure is pre-given. I have no special conjecture to make about how that comes to be - it's all a logical structure, prior to matter or thought, so unlike physics, it's not like there could be another universe out there with different basic mathematics.
It seems to me that these are two different claims. I think mathematics is a human construction and doesn't have a real substance. Instead, mathematics is a system of assumptions and generative rules, and more generally a discipline around creating and operating such systems. But "truth" within a system of assumptions and generative rules is not subjective, it's mechanically provable.
A possible confusion remains in what "true" means. Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality? I think it's trivially true. Gollum was a hobbit, within the system of Middle Earth. If it's not true, then we need a different word for true that does mean this, because this is what most people mean when they use the word about all sorts of imaginary constructs, from institutions to cultural symbols.
This reflects the development of intuitionism. The project was first started by L. E. J. Brouwer, who rejected formalism. It was then developed by his student Arend Heyting, who formalized it with the Heyting algebra, a restricted variant of Boolean algebra that lacks the double negation elimination (~~p => p) and law of the excluded middle (p OR ~p).
Pretty much all work in intuitionistic mathematics continues the work of Heyting. Brouwer would have rejected the entire enterprise as subjective, so he is mainly of historical and philosophical interest.
This is essentially the standard mathematical approach developed during the early last century. From a few basic axioms (which are not really justifiable) new statements are proven and structures are built up. All notions of truth and provability are relative to that system. In standard ZFC (those are the standard axioms) mathematics "1+1=2" is, like all other statements, a statement about sets. The statement is true by the definitions of 1,2, "=" and "+". In an alternative system with different axioms or definitions the statement is false.
This is not the view of intuitionists though. For them the symbols 1,2 or + aren't formalized objects (e.g. sets in this case). They are just symbols, which transfer some (hopefully) shared meaning to another person, who then (potentially with some addition arguments) might also accept the truth of that statement.
In the former the question of "truth" is fully formal there can be no "interpretation" in any meaningful sense. Intuitionism places mathematics fully inside the mind of the mathematician and "truth" can only be found there.
>Can something relating to an imaginary system can be true, or is it false because truth can only apply to objective reality?
In formalized mathematics "truth" is fully formal. There can not be any "external" truth derived from it, as any such statement is non-sensical.
Mathematics is a human construction, but it certainly has a real substance. What does "mechanically provable" even mean, if there is no absolute truth? Do you believe in the definition of a proof or not? Do you believe that whenever the assumptions of your theorem are true, and you have a proof of a conclusion, that then the conclusion is also true? If you do believe that, that's your absolute truth, then. If you don't believe that, a proof is meaningless, isn't it?
Different Axioms lead to different provable statements.
Believing in standard mathematics basically means that you can not believe in absolute truth. Unless you also believe that some guys a hundred years ago figured the sole and completely perfect rules which totally correspond to reality.
I agree that if one follows an axiomatic approach strictly and consider "truth" to be a shorthand for "provable from in some logic from some set of non-logical axioms" [1] then one is rejecting any notion absolute truth, since everything is relative to some set of axioms, but I don't agree with the charactedisation of this as "standard"; it seems to me to be a very Formalist stance.
I'd argue that most mathenaticians consider themselves Platonists, and believe that the mathematical objects they are describing are real enough to form some kind of metamathematical "standard model", and "absolute truth" can be defined in the model-theoretic sense relative to this standard model, even if this is somewhat unavoidably handwavy.
[1] : Even if you do think this, "truth" is generally used by logicians in the model-theoretic sense of "truth in some specific model/structure compatible with the language".
I used it as an objective term, defining mathematical objects on terms of ZFC and truth being relative only to ZFC is the standard mathematical foundation. If you ask a random mathematician what he thinks the foundations of mathematics are it will most likely be ZFC, even if he disagrees with it on any level, it is still what he would set his rival theory against.
As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
I don’t think it’s contradictory to work in ZFC whilst simultaneously having a non-axiomatic notion of mathematical truth.
I would hazard a guess that most (all?) working research mathematicians would accept the truth of the Gödel sentence for their preferred axiom system (and deductive calculus), be it ZF/ZFC or TG or something else entirely, so I cannot accept the claim that they see the “standard” notion of truth as being relative to all models of some axiom system.
You might think this is just nit-picking, but if it’s fine (in the sense that this is still “standard”) to add an arbitrarily large set of Pi_1 formulae to ZFC from repetitions of Gödel 1, then I don’t think we can say that ZFC is the standard basis of mathematical truth because this cannot be justified in ZFC; there must be some other (standard) notion of mathematical truth used to justify this.
Which is why I called it "standard".
> I don’t think we can say that ZFC is the standard basis of mathematical truth
You literally just said that working in ZFC is "standard". A standard is a social agreement, standards can be completely false and absurds, while being standards.
>As a sibling comment mentioned, most mathematicians have a sense of truth that is not bound to any axiom system.
Which I agree with.
> truth being relative only to ZFC is the standard mathematical foundation
I think this is at odds with:
> most mathematicians have a sense of truth that is not bound to any axiom system
Re.:
> You literally just said that working in ZFC is "standard"
Nowhere did I say that considering ZFC to be the arbiter of mathematical truth is standard, in fact I’m claiming the opposite.
Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever.
Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
I don't think so. Mathenatical statements are almost always framed in the context of ZFC. This does not contradict that mathematicians think ZFC is not an absolute truth.
>Perhaps I’m misinterpreting your claim? At the moment I’m reading it as: “The belief of the majority of working research mathematicians is that mathematical truth is defined relative to ZFC.”
All I am saying is that mathematicians are framing their results in the context of ZFC and that this makes it the "standard" theory. I think that statement is absolutely not controversial, even alternative theories are framed in opposition to ZFC.
I absolutely do not think that mathematicians believe that ZFC is "true", as in it is the one and only perfect set of axioms.
My initial argument (and I am sorry if that was unclear) was that IF you believe that mathematical truth is about formal derivations from axioms (ZFC would be such a theory, same as ZF or any variation) then either you have to say that there is one perfect system and all truth is relative to it alone or that there are multiple equally true, but incompatible, theories.
The "IF" is of course important and I don't think many mathematicians actually agree with the IF clause. I actually completely agree with: "Deciding to adopt a particular set of axioms as standard just means that I’ll accept a proof from those axioms without question; it doesn’t mean that I believe mathematical truth is precisely that which is a syntactic consequence of ZFC/TG/whatever." and I am sorry if I wasn't clear. I actually do not think there is any disagreement here.
Even your comment references “perfect rules which totally correspond to reality” which seems to be another way to say “absolute truth”.
I am certain about that, but it does not make my statement less true. I actually think that very few people believe in ZFC as either a formalist absolute or as an arbitrary set of rules. I think the most common view is that it enables other theories, that those mathematicians actually care about. The moment those theories rely directly on axioms things get difficult. I think the following quote describes quite well the state of ZFC: "The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
if you derive a different result, by soundness those would be equivalent ???
therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.
somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.
all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")
the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")
the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)
This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.
I insist that math is ideal. it models reality ideally.
this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something.
the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols is ideal.
i suppose what it all is all about is the intersection between this reality and this ideality.
For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?
We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.
Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".
We model the world and the world follows the model and the exceptions add to our refinement of the model and that’s what makes it true.
Perception and reality are yin and yang and that is about as deep as my philosophy needs to go here
Although it can be challenging, this exploration can also bring greater depth to our perception and reality, much like the yin and yang you mentioned. Even if we don't adopt a new philosophy wholeheartedly, engaging with it can open us up to valuable insights.
For example the toplevel article says things like "independent existence in an objective reality".. this sounds important, right? Perhaps depending on the truth of the intuitionism, we might discover some facts about real world?
Nope. The whole "is intuitionism true" question only affects the person thinking about it. The whole books are written that basically quibble about dictionary definitions of "fundamental principles", "objective reality", "mental activity" and other complex words.
(This is especially visible in discussion of "consciousness": there are tons of texts about it, and yet none of them matter in any practical way. The practical applications -- generative text models, NLP, neuroscience, etc.. -- just ignore all the philosophic cruft)
It's now widely known that an intuitionistic proof has computational content (through the curry-howard correspondence)
Those things have a real world impact
For example, Robert Goddard, who invented first liquid-fueled rocket, became interested in space when he read Wells' "The War of the Worlds". So did Wells' work had an impact on rocket science? Certainly. Does it mean you should read "The War of the Worlds" if you want to build rockets? Probably not.
It's like knowing design patterns in OO: it's ultimately all software right? But software exists first in the mind of the programmer, and what's written in the IDE is just a shadow of what the programmer imagined. This makes one appreciate the point that there are things in software that exists entirely inside our heads
Studying history will very quickly dissuade you from this notion, over and over again. For the perhaps most striking example, consider how different the 20th century would have looked like if a certain philosopher named Marx hadn't tried to marry the works of another philosopher, Hegel, with economic theory.
(It's an interesting thought experiment what would have happened if there were no Karl Marx. February Revolution would have happened anyway.. but would bolsheviks still exist? Would Lenin still do all the things he did anyway, or would he become a civil servant instead?)
And for the most striking example, consider how different the 20th century would have looked if Academy of Fine Arts Vienna would have accepted a certain applicant from Linz. Does this mean fine arts educational institutions are very important in the political life?
Sure, given the economic and political realities of Russia at that time, something might have happened even if Marx hadn't existed, but it might have looked very different.
Philosophy matters to the world because other people read philosophy and are inspired by it to change something about the world. The French Revolution, US independence, Latin American independence and so on were all undertaken in no small part by people who had read a lot and were trying to make some of their ideals reality.
In the philosophy of mathematics, intuitionism
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
When we say (or think) "Joe and Sue went to the grocery", it seems inherent to this statement that there are two distinct actors. Joe and Sue. But that is already math, isn't it? I have the feeling to avoid math, we would have to avoid "something" in the first place. As it already implies that "something" is in a category, consists of a collection of other things etc. So we could not talk about anything anymore.
That statements seems very obviously true.
remove all blackboards and chalk, and paper and pencils.
can you still do math?
Since the mind and its mental constructs are a part of the objective reality, they will end up describing aspects of objective reality. If they don't, they break down, become chaotic and incomprehensible to those grounded in the objective reality.
I’m no physicist, but I understand that Newtonian physics aren’t strictly true as such, but they are a good enough analogy to put a person on a different planetary body.
So I think it’s fine to be agnostic and practical about the outcomes without needing say much about the metaphysics either way.
Anyway, both perspectives tickle my curiosity.
I'm of the "all models are wrong, some models are useful" school of thought. My best guess is that the platonic-ideals-are-real folks are mistaking something in their head for something outside it. That's not to deny that there is an objective reality out there, just that I have no particular reason to think that it's perfectly representable in 3 pounds of primate headmeat and expressible by squirting air through our meat-flaps. [1]
But ultimately, it doesn't matter too much to me, because the practical utility of both models is pretty high. It does make me wish to meet intelligent beings from different evolutionary backgrounds, though, as I think there would be a lot of "So you think what exactly?" that would be very revealing about which things are pan-human quirks and which are more universal.
[1] Credit goes to Terry Bisson here for the last bit: https://www.mit.edu/people/dpolicar/writing/prose/text/think...
Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature, they'd be scientific questions if they did. The goal is to massage the question into an answerable one if you can. It's not really possible to have a "wrong" answer if the question is unknowable or malformed.
I could have made the original post with less melodrama but I like to snub philosophers when they swoop in like they have answers to these questions. Either it's a matter of science or you don't have answers, just perspectives.
I think a form of logicism (the view that mathematical statements are logical tautologies in disguise) is true at least for arithmetic. There are some ways to interpret them using logical truths from pure higher-order logic. I'm not sure about most other parts of mathematics.
> Here's one of mine so I play by my own rules: philosophical questions don't have any testable hypothesis by nature
I think they are testable by performing conceptual analysis, which consists in organizing data from semantic intuitions about the involved concepts.
"Socrates is a male" has a value of True; "All men are mortal" has a value of True; and a conclusion of "Socrates is mortal" that is "True" because a True "and" a True yields a True.
And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
This causes "Socrates is mortal" to have a truth value of "False", which doesn't make sense unless you consider it "proof" instead of "truth": it used to be proven (by the premises) that Socrates is mortal, but now it is "false that it is proven". It might still be true, but it's not proven.
This is closer to intuitionist logic than classical logic, but it still relies on facts like "Socrates is male".
No, for a logical argument the conclusion must be true if the premises are true. A valid argument (proof) only shows: necessarily, if the premises are true, the conclusion is true.
> And then an immortal man is discovered, making that first premise "False". And False "and" a True yields a "False".
> This causes "Socrates is mortal" to have a truth value of "False"
No...
And via Curry-Howard, any intuitionistic proof is also a computer program. Intuitionism thus unifies computation and mathematics in a very direct way, which has been extremely useful.
The Metamath system lets you specify the axioms you want to use, and then can verify that your proofs only use those axioms (directly or indirectly). There's a Metamath database specifically for intuitionistic logic:
https://us.metamath.org/ileuni/mmil.html
More things have been proven using intuitionistic logic over time.
That doesn't mean the external physical world doesn't exist, the information we use to construct that model must come from somewhere, and we can deduce that the source is a persistent and consistent one.
The philosopher Husserl said: “The tree plain and simple, the thing of nature, is as different as it can be from this perceived tree as such, which as perceptual meaning belongs to the perception, and that inseparably.”
He came up with the idea of the noema which is our experience of something, and noesis which is our conscious act of perception. For me, that's our act of interpretation of our sensory perceptions. Sometimes this all goes wrong and we construct a flawed model that does not correspond perfectly to actual external reality, such as when we are deceived by optical illusions, stage magic or just hallucinate. Fortunately we can test and correct our perceptions through action in the physical world.
I'm an out-and-out physicalist but I think he is quite correct, we must distinguish between our internal perception of things and how things actually are. Fortunately science is extremely powerful in this regard. It has allowed us to decouple our model of the world from the limitations of our perceptual system, and come up with rigorous models of reality such as Relativity and Quantum Mechanics that are not tied to direct interpretation by our perceptive systems.
When we look at a tree, it could very well be that the source of the tree is very much like the tree we experience, but it could also be wildly different. When we see a tree in a video game, we know there is no real source tree just like it, just ones and zeroes. I disagree that science fixes this problem. Tools are still just measuring the physical world. For example, if you used a tool to measure some aspect of the tree, you are still measuring the representation of the tree in this world. If I use the video game analogy again, my point is that you wouldn't be able to see true underlying 'source code' of the game tree by looking at it in the game.
I don’t expect any description to accord perfectly with the reality of the object it refers to. It’s just a description, which may be more or less accurate or useful. Science, and investigation in general, is a way to test and improve such descriptions.
Well, of course, they kind of can. There are drugs that make the world look like it's squashed, such as ketamine. The way that we perceive reality really is totally dependent on the properties of the observer. Of course we all, with our sober minds, assert that we are perceiving the ruler the "right way", but all this means it that we perceive the ruler in a way that most humans agree with. Jumping from that to "this is the way that the ruler looks for all possible subjects" is a leap of faith.
When you take a picture of an apple, you have a picture of an apple, not the apple itself. Both are real and related, but not the same thing.
This gives an argument about intuitionism: If you say that math is a byproduct of our wetware and nothing else, how come we can successfully teach it to turing machines, and have that process fill us on some holes we had in our understanding of maths, but not terribly large holes?
Too many people believe in the law of excluded middle, especially in their own lives, much to the folly of civilization itself.
If you believe that "now" is real, you believe in LEM.
The neat thing about this is that in principle it could be tracked in a proof assistant with type inference. The ZIO framework for Scala has a super slick system for dependencies where e.g. an `RIO[Foo,A]` (an IO that requires a Foo and returns an A) and `A => RIO[Bar,B]` can compose to form an RIO[Foo&Bar, B] with the types inferred. So you could in principle have a proof system that lets you infer types like `A => Reader[LEM[Foo] & LEM[Bar], B]` (where LEM[T] = Not[Not[T]] => T), i.e. you get explicit types that show all of the instances of an LEM function you need to make your proof constructive, and since you're thinking of things as living in the Reader monad, you have the ergonomics of a proof that just assumes LEM.
Same idea could be used with AC or any other "controversial" axiom. So you don't need to "believe" them to use them, and if you do "believe" them, you can benefit from pretending you don't.
"In theory, mathematics is all pure & abstract logic, with no connection whatsoever to the real world.
"In practice, if you want funding for your mathematical research, or for more than a puny handful of people to ever look at what you did, then you had best pay plenty of attention to the real-world usefulness of it."
I understand what your math professor was trying to communicate, but at the same time, I think framing things this way begs the question that mathematical entities are not "real". It's more accurate (or at least less question begging) to say something like "mathematics, or mathematical objects, doesn't seem to have any obvious or direct connection to the material or physical world". Putting things this way doesn't reify mathematical entities, but it also doesn't presume that they don't exist.
In the context of the Godel Incompleteness Theorem I have always found it difficult to reconcile mathematics w/ an objective reality. A mathematical system would simply be incapable of completely encompassing objective reality.
I guess the opposite would be something like solipsism though.
Although I understand it’s unlikely philosophers will ever agree on what reality is.
If anyone feels compelled by this you should probably start by learning about the many rock-solid antipsychologistic arguments out there that are at least 130 years old:
In the same way that psychology can be harmful to the pursuit of truth, there is an evil or deformity in what pure mathematics represents (or became), as Cantor and Gödel both discovered trying to eliminate its paradoxes. Mathematics as practiced by, e.g., Leibniz and Euler, was much more interested in calculation, i.e., mechanization. It is this form of "mathematics", as the thing has come to be called, that should have been developed, instead of trying to axiomatize infinite sets, explain the continuum, or fix impredicativity.
But despite the many paradoxes of mathematics (e.g., AoC, LEM) and machines (i.e., bugs), computer science has happily chugged along and made many practical contributions which explain the universe and shed light into people's minds, fulfilling the role that pure mathematics once served. Today, we are very close to mechanizing both minds and mathematics, and can replicate many aspects of reality (i.e., mathematical or otherwise) inside computers. It is not unimaginable that one day, computer science as it is practiced today will be seen as a kind of telescope for revealing the nature of mathematics and/or reality, which the mind alone cannot fathom.
My personal problem with math that prevents me from seeing it as "discovery of fundamental principles claimed to exist in an objective reality" is natural numbers. It's impossible for me to clearly find the number 1 (for example) anywhere. The boundaries between one unit and 0.X or 1.Y seem arbitrary and chosen to help us create models. Religion and philosophy deal with this, but it's also apparent in digital signal processing. Or medicine: are the numerous bacteria in our bodies us, or not? Is the heat radiation from my body me? What about the fact that 1 + 1 rarely if ever equals 2? Meaning that two things together physically interact to create more than the sum of both parts. From celestial bodies to a replicated database the complexity goes through the roof when you have more than one thing.
There is zero evidence ZFC is inconsistent.
Even if "1" does not exist in reality mathematics still describes fundamental universal principles. As long as you believe that these fundamental principles exist at all they exist as mathematical ones.
Not even hardcore Platonists would claim that the number 1 exists in physical reality. But that does not mean it doesn't exist in some abtract sense. You can construct Models of reality using the natural numbers and these models about real objects are just imperfect descriptions of reality.
ZFC is weird (especially choice). It's not implausible, but there's little a priori reason to assume that it describes some phyiscal reality. It just happens to give a foundation to a lot of really useful mathematics.
You could take a theory such as Peano Arithmetic and argue that that one is self-evident. But unfortunately, again by the second theorem, you can't use PA to prove ZFC consistent. That's, roughly, what Hilbert wanted to do in order to convince his critics, and he failed.
[0] https://projecteuclid.org/journals/notre-dame-journal-of-for...
Inconsistent: trivial, from falsehood follows anything.
Incomplete: Peano.
Weak: Presburger.
Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.
Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...
No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.
what about that there is one of you?
I have exactly one wife. If she gained weight, I would not then have 1.01 wives.
I have exactly one cat. If she had N kittens, I would then have exactly N+1 cats, not (1 * N/10) cats.
And so on.
Once the grey blob of infinite nothingness becomes distinct enough that there is a difference between one moment/point and another moment/point, you have enough to start counting different states (or do binary arithmetic). Its high school math to find out some numbers are different then others and eventually you'll find the primes.
I'll bet there is no god, power, or alternative rules in any possible universe, fictional or real, that could not find the primes.
But, like, just in someone's mind ... to keep the truth of that existence purely subjective.
You may have noticed that a lot of the cliché philosophical questions haven't really shifted for as long as people have been asking them. It isn't because there is no answer, but rather the question itself is null and void. It's answer is beyond our ability to find or compute because you would have remove yourself from the human frame of reference to find it.
The meaning of life? Doesn't make sense outside of the context of human life. In that context, it's whatever you want it to be or decide it is to you. Outside, well, you'd have to die to see if there's anything beyond, and as Spock points out in ST4, it would be impossible to discuss without a common frame of reference with someone who hasn't died.
~~~
To claim that math is a human invention or a noumenal language of nature is futile, the answer is not within our ability to determine. It is, for all intents and purposes, undefined. Unknowable.
That's not to say that the conversation is meaningless, as the logical positivists would claim, as this is also a leap too far.
Early Wittgenstein, big influence on the logical positivists, didn't try to claim that philosophy was meaningless, just that if you try to talk beyond the facts, you won't get anywhere.
Bradley aways summed it up nicely for me: "anyone ready to dispense with metaphysics is a brother metaphysician with a rival theory of his own"
In other words, to rule out metaphysics (defined as concepts and ideas that lack empirical data or basis in fact) is to make a metaphysical proposition, since by definition, there is no data to disprove the metaphysical propositions, just as much as there is no data to validate them.
Most metaphysical philosophies caught on because they can be presented rationally as a chain of propositions and conclusions, but they only add up within their own framework that is upheld by an unprovable assumption or set of assumptions.
Example, cogito ergo sum makes perfect sense, provided that you accept that the speaker is I, that the speaker thinks rather than simply utters, and that to be means anything at all. In the end, cogito ergo sum can be accepted intuitively within the framework of "let's not be sceptical of every thing" but what can you conclude from it?
"I think therefore I am" well, I is a thing that thinks
-> "I thinks, therefore it exists" I think presupposes I's existence
-> "I thinks"
That is a definition of I
I = thinks === I is a consciousness
which is a tautology. And that's why is makes sense, because it doesn't go anywhere than where it started, it's algebra.
It starts off as a = b where b = a and can be reduced to simply a
Math is privileged in that its algebraic statements can be used to model things in the world because it's numbers and functions of numbers, but ultimately the usefulness emerges from proving that a equals a very complicated statement that isn't obviously tautologically equal, or not equal to, a.
Math is a tool. It's a way of reasoning that comes bundled with reasonably standardised notation that enables boosted productivity compared to reasoning about the same problems in regular languages.
~~~
It's the same reason why I find "simulated reality" questions rather dull. If the world is a perfect simulation, we have no way to distinguish it from the real thing, and so to our frame of reference, there is no difference about which we can have a discussion that makes any sense.
If there are cracks in the simulation, sure, then you have a fact to talk about. But then the discussion isn't philosophical, it's practical: We've been brain-hacked, what do we do? and the conversation ends rather abruptly.
I prefer to think of mathematics in general as a language, we are constructing descriptions of relationships between concepts, and hopefully those descriptions turn out to be consistent ones. When a description is proved to be consistent, we say that it is true. I suppose that makes me an intuitionist perhaps?
Everything I said in the first paragraph above about mathematics also applies to descriptions in any language. There is the abstract set of all possible statements in English. There is the set of statements that have been made, or at least that exist in writing, recordings and people's minds and therefore exist encoded physically. There are statements that are grammatically correct or accord with linguistic conventions, and also those not unlike what the appearance sensibly is or may not be. There are also statements that correspond with reality, such as a biography of a real person, and ones do not like The Lord of the Rings.
I think of these things in terms of physically encoded information. There is the collection of hypothetical information that could exist such as plays Shakespeare never wrote, and the subset of information that does because it exists in a physically encoded form. I'm not a Platonist, what we call a circle is a description of a geometric form, and a real geometric form is a circle to the extent that it matches that description. There is no abstract form called circles that exists in any sense, or any world of forms for them to exist in. Actually I don't think Plato thought there was either but he didn't have a robust account of information to work with.
Taking this to the relationship with science, there are many, many valid and consistent mathematical formulae, descriptions, theorems, etc that are proven consistent but have no correspondence with anything in physical reality. No process, no physical structure that they describe. These are like literary fictions describing a fantasy world, although they may be mathematically rigorous. However there are some mathematical descriptions that do accurately correspond to relationships and processes in the physical world, and we can use them to predict the behaviours of those physical processes. This is because, fortunately, the physical processes occurring in the world are highly consistent and persistent, and therefore can be described in a highly consistent formal language such as mathematics. We call those physical laws, though I hate the term laws. They are simply highly accurate and predictive descriptions of behaviour we have observed.