Is Math Real? | Hacker News Reader

Is Math Real?

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208 pointshhs336 comments

> one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively

I havent read the book but this dichotomy is not an intrinsic feature of how people pursued mathematics historically.

The "stiffness" and excess focus on rigor and accuracy developed gradually over the 19th century because people were being loose canons - primarily around calculus.

This is going to be a classic "computer programmer wants math to be like computers and doesn't get it" kind of take, but In my view the problem of mathematical rigour (or lack thereof) has only become worse over the 20th century and we certainly did not resolve any of the underlying issues in a foundational sense or a practical sense. In a practical sense, it's become much worse and we have many more layers now. In a foundational sense, we succeeded in giving up because we learned that we can in some sense pick and choose whatever is most convenient for our line of research.

That's probably okay if we view mathematics in the way this book (I have not read it, going based on the description here) advocates, as a sort of toy for playing with arguments. And I'm certainly not saying Math should ever be viewed as an empirical discipline nor constrained by that kind of thinking. But I don't think I'm the only one that takes one look at things in the realm of say higher category theory and thinks it's mostly playing word and symbol manipulation games, and lacks any real mathematical content that could not be discovered at a lower and more understandable (and less likely to produce new research) level of abstraction.

I guess I've sort of betrayed that I am pretty firmly a platonist in that respect so make of that what you will.

Like I said, this is not an unusual opinion for a computer person to have and I'm sure it's fairly annoying to any pure mathematician at this point. But I think it's still fair if we want to understand what turns certain people off of pursuing mathematics further.

I'm not a mathematician so anyone please correct me if I'm wrong, but isn't the point of category theory to provide abstractions so your proofs are very general and cover a lot at once? Of course, it needs to be very abstract to achieve that.

This is a classical example of what happens when people give up on rigour:

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

TL;DR: They started producing false results.

Note that this was not about capital-F Foundations of Mathematics like (arguably) the foundational crisis of math that had its origins in the 19th century, but rather about lowercase-f foundations of a particular field, in this case algebraic geometry.

Weil's foundations of AG in the eponymous 1946 book were horrible and messy but they solved the issue (even today there are a lot of celebrated results that can only be found as expressed in Weil's language) and later in the 1960s Grothendieck provided the elegant language of schemes in which people generally learn and research AG today, and which helped prove long-standing problems like the Weil conjectures and (to some degree) FLT. Category theory was, in this case, essential to proving theorems about "real mathematical content" like numbers and points.

Is that really the case? I'm not an expert in AG, but I would imagine that any major result published in Weil's framework can either be trivially converted to schemes or it has been ported over by now

IMHO, in order to be able to speak about these subjects, we must either be a mathematician, or have read logicomix (http://www.logicomix.com). It is the easiest book to read. Another usefull one is https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

This subject actually doesn't belong in mathematics: it's philosophy of mathematics. I.e. no matter how much you study mathematics you will not be able to answer (or even attempt to answer) these questions because no mathematical tools or disciplines are designed for that.

I also believe that the emphasis on "rigor" here is misplaced. The argument isn't about whether mathematical rules are rigorous or not. The argument is about whether mathematics exists independent of mathematicians (and they discover it in a way how an astronomer peers into telescope and discovers new stars) vs mathematics being created by mathematicians' minds (similar to how an architect designs a building: there weren't one before, and now there's a concept of a new building with so many walls, floors, windows etc.)

I believe that mathematics is art, not science. I.e. mathematicians create new rules, they don't discover them. The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?

> The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?

The "surprising effectiveness" is, I think, one of three things. First, the surprise is that we can create mathematics that describes our physical world.

The second surprise is that, when we find out something new, it often takes the form of existing mathematics that we didn't design to describe the physical world. (Though, from your point of view, I suppose you could say that we created mathematics to describe everything that could be described by mathematics, and so it's not surprising that something was there to describe reality.)

The third surprise (maybe this is just a restatement of the first one) is that mathematics really describes the physical world. It's not that we find some math that describes it, and then we change the situation a little bit and we need to find some new math. The surprise is that the math describes what is going on so well that it applies to situations that we didn't know about when we devised the math that applied. That is, it's predictive, not just descriptive.

Well, the last one is a kind of surprise like... if you believe in real numbers, then there are some nonrepeating real numbers (pi is believed to be one of them) that embed all other real numbers. I mean, to someone unfamiliar with the concept it sound amazing and improbable that some number embeds all other numbers (all of which are infinite). But this is because we aren't used to dealing with infinite things.

Similarly, if mathematics is so powerful as to be able to describe any physical reality, it shouldn't be surprising that it can describe ours, no matter how complex and detailed.

All ideas are discoveries, not creations. The fact that anybody else can come up with the same idea lends credence to this. That's how ideas work - they are transcendental.

Your argument is similar in spirit to the claim that all, even infinite sets can have single element selected from to form a new set (so called "axiom of choice"). I.e. you equate the mere possibility of existence with a proof of existence (it's not a slur, there are a lot of mathematicians who do just that).

I don't see the reason to think that coincidentally arrived at ideas mean that ideas are discovered. We use the same framework, with the same rules. It's not unlikely that we'll create same ideas, because we use all the same rules, but the framework is so vast... it has so many rule combinations it's mind-boggling how would you think all ideas already exist in this framework.

More practically, we call an action a "discovery" when we (unbeknownst to us) faced the consequences of the phenomenon being discovered, but didn't know why we were facing them. An astronomer who finds a "new" star was ever so slightly influenced by that star's light, gravity etc. A marine biologist who discovers a new deep-water fish was ever so slightly affected by that fish through a complicated chain linking many different species through biosphere.

An artist drawing a new painting isn't discovering it in the same sense. She isn't interested in how an existing painting was connected to the consequences of her life or the lives of the whole human species. By selecting of all possible ways the paint can be laid down on canvas her particular way of doing that she creates something genuinely new, or as new as it can possibly be. It's counterproductive to label this activity "discovery" because then we lose an important distinction between the nature of the work of an astronomer and that of a painter (or a mathematician).

There are sculptors who'd jokingly say that they "discover" the statue in a stone slab. But they do so in a sarcastic kind of way, really (well, artists are weird and will make a lot of nonsense claims just to trigger non-artistic audience). But, deep down, nobody believes that they are searching and finding good images, melodies or novels. I cannot really imagine a mechanism through which I'd discover the answer writing to you. It's a lot easier to explain what i wrote by saying that I meant to write it.

IMHO, the book was too loose with the truth (even though it was purporting to write about mathematical truths). It's a work of fiction (beyond just filling the characters' words to each other), but that wasn't revealed until the end which left me sour. It pretending to be history right up until the end

GED isn't really related to this subject although it contains some of the same themes and characters

Its still not stiff enough. Mathematicians really ought to settle on a sound and coordinated syntax

Syntax is about the least of people's worries when they worry about 'mathematical rigor'.

Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math.

Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another.

That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics.

My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.

While I agree with the point you make, I think that syntax has a huge impact. Maybe not on mathematicians themselves, but I am certain that mathematical syntax has done more to scare people away from the field than the actual mathematical problems themselves.

For example the convention to use the greek alphabet for certain things. This is totally arbitrary and you could have also used emoticons instead (had they existed). But what this means is that the pupil, before tackling the meat of the mathematical problem has to accept that weird looking letter they have never seen for no real reason whatsoever.

And I say that as someone who can fluently read the greek alphabet.

It is not arbitrary, it is heritage. Apart from the fact that Greek was the de facto scientific language of the west (which is no longer the case), I think we can agree that the characters of a single alphabet are not enough for notation, especially given the fact that it is very useful to be able to discern different types of entities, e.g. constants from variables or vectors from matrices.

If we changed symbols now, it would create an even bigger mess. Because the people that learn the new symbols, could not read any textbook published before 0 A.D. (Anno Discombobuli)

There are simple practical reasons though. We don't do that to be fancy. There are simply not enough letters in the latin alphabet to not have common intersection in writing. We like to use the same letters for objects of the same type (like x,y for coordinates or i,j,k,l for indices) because that increases readability significantly. But it does mean that you run out quite quickly.

Adding another alphabet alleviates those issues somewhat but even with greek letters added in we still run into this issue somewhat commonly.

I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?

Mixing up 'f' and 'f(x)' is mostly harmless in practice. The underlying principles are still clear enough. (And I say that as someone who would _really_ like to make that argument that people who mess this up are somehow unclear in their thinking. No, they are mostly fine.

Getting 'f' vs 'f(x)' right mostly is really important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category.

You could say calculus deals with higher order functions, like the derivative. And that's a valid way to look at it. But most people get by just fine using special purpose notation for the derivative and not thinking about it as a function just like 'f'.)

I used := to emphasis that I am defining 'f' here, not just writing down any old equation. (Eg like like in the example "Find all functions f such that f(x + 1) = x * f (x).")

Though if you wanted to be pedantic about notation, I could have written that as with the x on the other side of the :=, like f := \x -> x + 3 (for Haskell inspired notation) or f := (x |-> x + 3) where |-> means the little arrow I draw by hand to denote a mapping when I'm writing math on a chalk board or piece of paper.

I'm not sure why := would denote a computation? At most you might want to use it to denote an assignment in a mutable context?

> Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?

In math, := is typically used to denote a definition. Using equality (=) only makes sense if both sides of the equality sign already have a definition.

I agree but i'm by no means even remotely an expert. Mixing up f and f(x) seems pretty bad to me.

f = y+3 makes sense f(x) = y+3 does not make sense (at least to me), f(y) = y+3 makes sense however.

f(x) is a function of x correct? It's articulated as "f of x".

> I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?

:= is assignment in Pascal iirc, maybe that's where it's coming from.

Syntax and notation is an incredibly minor "problem". We have much more significant (and interesting) stuff to deal with.

Judging maths on its syntax is like judging a poem or work of literature on its font. It really isn't a central thing.

This book is not really addressing the more common "is math real" question of it being empirical or invented. For an interesting take on that question, see the 1st section of the 2nd part of Daniel Shanks' Solved and Unsolved Problems in Number Theory. He makes some interesting points about the old Pythagorean views

For me, both questions "is math real" and "is math discovered or invented" miss the point. Math is a model of the universe in the same sense that a world map is a model of the earth.

Is a map real? Well, it is. I can see it on my desk. Is the earth real? It is too, but they are not the same. In that sense map is also not "real".

Is the map discovered? Well, it uses data that was mostly discovered, but some parts were "invented" or edited for simplification for the map to be useful.

The real question should be "is math useful" as a model. We all know most basic parts are, but some mathematicians forget that they are dealing with an imperfect model and keep finding paradoxes. It's like we would forget the imperfections caused by the mercator projection and be surprised the real world distances are not proportional to map distances.

That's the reason I always liked engineering more than maths. When programming you always "import" the libraries you need and find useful for the task. You only make sure that they are compatible with each other. Mathematicians "import" all axioms, call them maths, and are surprised they get paradoxes.

You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".

Math is nothing like a map -- maps are approximations of something real and they don't have any kind of internal consistency or complexity.

But there's a good argument that math is the fundamental nature of the universe, and mathematical discoveries lead to predictions of real-world behavior. While maps don't predict a thing.

The philosophical discussion isn't around whether math is useful for tracing the arc of a ball in the air, for which it always will be merely a useful approximation. It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.

And here, the philosophical questions around whether our descriptions of quantum physics are "invented" or "discovered" go quite deep, and necessarily involve the nature of human knowledge itself. For many people, these don't "miss the point" at all -- they're some of the deepest, most profoundly meaningful questions that exist.

> You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".

I read my comment again and I was surprised, as I did not intend this tone. I’m sorry for being dismissive and for generalising too much about mathematicians.

Could you elaborate or point me to a formulation of the “language of the universe” argument you mentioned that avoids mentioning quantum physics? I don’t understand quantum physics and I’d like to avoid falling for the quantum physics fallacy [1]

[1]: https://www.logicallyfallacious.com/logicalfallacies/Quantum...

> It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.

Why is it that everyone thinks of mathematical models of quantum mechanics as much closer to the "nature of reality" than any other mathematical model? If anything the constant disagreements between quantum mechanics and physical models at other scales should make it clear that all the models we have are wrong by virtue of incompatibility.

I think the question should go even deeper. There are so many fundamental axioms that must be accepted on faith alone. The question I usually start with is "Can anyone prove that numbers exist outside of our imagination?" I not talking simply about perception. Even I believe that if I perceive that I am hit with a brick then the brick exists. We have no senses that can detect numbers. When I asked this question to any of the several mathematicians that I know, the answer has always been ~ Yeah, good question ~ and then they move on.

Feynman has some useful words about this phenomenon of always wanting to dig deeper: https://youtu.be/36GT2zI8lVA?si=Boiqod3GXHVMyE_s

Why do you think we should go deeper with pointless questions? What would you do with the answer if someone provided one?

> "Can anyone prove that numbers exist outside of our imagination?"

What do you mean by "exist" here?

> Math is a model of the universe in the same sense that a world map is a model of the earth.

Except math can hypothetically model any consistent universe, not just our universe, which kind of undercuts the argument that it uses data that was mostly discovered, or that it's merely a model.

I think the most general view is that math is the study of structure, and some structures are real (in the sense that they exist in our universe), and some are not but we can still "discover" them by selective permutation or enumeration of axioms.

Cartography can also model any consistent universe, and I fail to see how that changes anything for the “it’s just a model” argument.

We can permutate and enumerate symbols for mountains, rivers and roads on a piece of paper. Maybe we would even get some “interesting” results like a map of the Lords of the Rings universe. How would that change anything?

> Math is a model of the universe in the same sense that a world map is a model of the earth.

That describes pre-1900s math we inherited from the greeks. With advent of non-euclidean geometry and abstract math, math is no longer bound to objective 'reality'.

>Math is a model of the universe

And I would dare to disagree right here. Math contains many structures that we don't know from our universe and that probably do not exist in our universe. If math is a model of universe, why is there a Mandelbrot set?

Yes, that is an intrinsic property of all models. They are imperfect, and we accept it as long as the models are useful for some purposes.

My map has a text written on it saying “Pacific Ocean”, yet I would not complain if I went to this place an couldn’t find a giant object in the ocean that would look like a letter P from the skies.

> This book is not really addressing the more common "is math real" question of it being empirical or invented.

Please note, this is mentioned at the beginning of the review:

"I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively.

The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”

Math is invented with a purpose to help humans understand the economy and the world around them. Economy, in turn, was also invented to help humans organize their resource use.

The more humans understood the world, the more they tried to apply math and other sciences (also invented by humans) in order to explain it.

It's not even a question. Two apples will always be two apples. It's just that, without math, it would be "an apple and another apple next to it".

Pythagorean ideas—- well, it has been some of the most enriching philosophy I’ve ever encountered. It’s a rabbit hole, for sure.

It's interesting that mathematicians, when asked philosophically, might have all kinds of interesting and nuanced ideas about this topic. But when you let them get back to their mathematics, they behave as if they believed deep down in their heart that mathematics exists independently of the observer.

(It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)

I am waiting for the universe to do something non-mathematical. There is no reason to believe it can't :-)

Sorry to break it to you, but most mathematicians don't give a s*t. Even worse, a large percentage of mathematicians (not sure if "most", but I'm afraid yes) do not usually have "interesting and nuanced ideas" (nor opinions) on anything.

This book is on my reading list. I listened to an interview of the author who said that their primary motivation of the book was less about teaching math, but teaching the value of looking at the world abstractly to determine patterns and constructs to explain our shared experience.

Is math real or not? It doesn't matter she posits, it works and continues to work and we can learn from its existence that almost everything can be explained and is "predictable" given enough inputs.

I don't know if it's real but it's definitely complex.

Came here looking for this comment, not disappointed.

OTOH it can also be rational and natural.

OTOOH sometimes it's irrational and transcedental so there's that.

And, amongst the reals, the irrational infinitely outnumber the wholesome

Get out.

A better question is - is math an inherent part of the universe or is it a uniquely human construct?

Math is a human construct to describe inherent parts of the universe (among other things).

The universe is compatible with math not because the math is part of it, but because the universe is what math was invented to describe. Most fundamentally, relationships between related structures and sizes. Obviously the universe is full of those since an order does emanate in ours. So aliens probably also have math and even discovering the same relationships etc but that still won't make math an inherent part of the universe to me. The universe doesn't care for math, it just is. Intelligent beings want to describe and discuss it though so we keep inventing math in order to do so and speak a common language.

Personally I find this very simple and not controversial at all. Math was simply invented as a system for us to teach and jot down things so that we don't lose knowledge across generations or for example colleagues.

Right, mathematics is a language for very formally and precisely describe consistent relationships. Since the universe is persistent and consistent, it is useful to us such descriptions.

I may not be possible to know why the universe is this particular way, but I don't think the universe is consistent 'because of maths'. The universe is this way for unknown reasons, but languages don't define or create the things they describe.

To prove this, we can construct descriptions of things that do not or cannot physically exist. Frodo the Hobbit, for example, in English. I'm sure there are equivalent expressions in maths that don't relate to physical things. The description, and therefore the concept exist (same thing), but the thing itself does not. Another way to say it is that the description does not correspond to something that is physically real. So we can construct mathematical descriptions of unreal or hypothetical things, and we can construct English language descriptions of such things. That's just a feature of languages.

The mathematical universe hypothesis says reality is mathematics. Not just that math can model reality, but that the universe is a mathematical structure.

That isn't really at odds with what you are saying though, it's just a lower level.

Mathematics is an inherent part of the Universe. What we Discover/Invent are Models/Notation for identifying and using them. The difference today is that we have built and extended the Abstractions/Relationships to such an extent that people feel it is not "Real" which is of course not the case.

The best argument for this are the various structures across the World (eg. Pyramids in Egypt/Central-Latin America, Temples/Structures in India, Aqueducts from ancient Rome, Great wall from ancient China etc.) spanning thousands of years which could not have been built without a knowledge of Mathematics as we define it today. Their approach and models/notation may have been different but the essence of the Mathematical Abstraction is the same.

Sometimes math can be pretty surprising, such as when complex numbers might be shown to be necessary to explain quantum mechanics:

https://www.nature.com/articles/s41586-021-04160-4

Or when group theory predicts subatomic particles through symmetries:

https://www.britannica.com/science/subatomic-particle/Hidden...

But other things like continuity and limits, as well as various topological spaces, seem to be purely mathematical constructs.

Continuity and limits, and some relevant topological spaces are absolutely required to describe very basic fundamental physics. You can't write down even stuff like Newton's laws without calculus, and you can't do calculus without something equivalent to limits.

See my other comment here: https://news.ycombinator.com/item?id=37656080

As the other commentator points out, "Continuity and Limits" are fundamental to explaining physical phenomena (via differential equations) and are not "purely mathematical constructs".

PS: You might find this interesting; Imaginary Numbers are Real - https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703...

There’s so much more math though. More than will ever be seen or used in structures, thought about, etc even in principle. E.g. if string theory isn’t real the math still is, so where is it? Part of the universe is pretty unsatisfying.

>There’s so much more math though.

True, that is why I used the words "Discover/Invent". Also i used physical structures as an example since they are the most visible and unarguable evidence of "Real" Mathematics from the earliest times.

You can invent abstractions to model concrete things(eg. all that is needed to model a skyscraper) or to model still further abstractions in a chain (eg. vector spaces for multidimensional/functional/etc. spaces). We only realize that it is "Real" when it is "Applied" in the concrete World (eg. number theory in cryptography).

Agree. Our math is convenient for us humans living in our universe.

But how much of our math is just a poor approximation of our universe? Like Newton's gravity was.

If our math only _approximates_ the world, if we discovered something that explains things better, it would all be irrelevant.

There are a lot of hints that something big is missing in our maths as a means of explanation. Like the mathematical constants Pi and Euler repeating infinitely, quantum randomness...

A good line of questioning is to explore the constants that arise in physics, of which there are nineteen[3].

E.g. "Why is the speed of light what it is?".

~300,000,000 meters per second. But the definition of a meter is actually defined by the speed of light, so this number is very human-math-specific.

So instead, you want to look at the speed of light in terms of other physical constants to find a "dimensionless" constant.

This leads us to the fine-structure constant[1], which is a single number that pops out when you relate a few of these experimentally measured constants to each other.

0.0072973525693 ≃ 1/137

This is a number that if any different would mean the universe would not exist in the way it does.

Something very human is the notion of "1". Counting things is very important to intelligent life.

I was thinking the other day, about the world from the perspective of a tree. It doesn't care about counting things. So "1" is irrelevant to it. It's an invented concept by humans.

And most of our mathematical thinking is based around this.

There could be an infinitely deeper and more complicated maths to explain things.

It's like looking at a leaf without a microscope to figure out biological processes. Until the 1600s, biologists could only study what their eyes could see.

All this quantum randomness feels like we are still just looking at a leaf with our eyes.

[1]: https://en.wikipedia.org/wiki/Fine-structure_constant

[2]: https://en.wikipedia.org/wiki/Dimensionless_physical_constan...

[3]: https://en.wikipedia.org/wiki/Physical_constant#Number_of_fu...

Plants may not have a concept of numbers or math, but they seem to follow mathematical patterns regardless.

https://thatsmaths.com/2014/06/05/sunflowers-and-fibonacci-m...

How do you know that trees don't care about counting things? Have you ever been a tree?

Would in a completely different universe with completely different physics where some life and civilization (for whatever that means in those different physics) manage to emerge, this civilization have similar mathematics, or completely different?

Is it possible for a universe to exist where those who can think and would follow all possible logic rules, find that e.g. the natural base of logarithms turns out to be something else than 2.71828, or get different prime numbers in the integers despite using similar addition and multiplication rules, or other such changes..., or would they find exactly the same?

I think they would find exactly the same (when it comes to the real actual logic, they may use different conventions and focus on different things if they got e.g. a different amount of dimensions in their universe etc...), I simply can't think how following logic rules could conclude something else no matter in what universe...

This depends a lot on how different you imagine this universe could be.

For example, in this different universe, if I have an apple and you give me another apple, how many apples do I have? If I have 2, just like in our own universe, then you're probably right. But what if I have 3 apples? What if I still have 1 apple?

We can certainly create number systems that don't behave like the integers, or addition operations where 1 + 1 = something other than 2. We haven't explored many of those too much because they're not very interesting, but they still have structure and may have similar concepts to what we call prime numbers etc. The integers and regular addition happen to be much more useful for understanding our world than all of these other systems.

In a vastly different universe, the opposite may happen: if they studied this weird operation where 1+1=2, they would reach the same conclusions as we do. But they never study it, because it doesn't match their universe at all.

> the natural base of logarithms turns out to be something else than 2.71828

The Euler number has a very concrete definition (or, actually, quite a few equivalent ones). The answer is clear if definitions are the same (all - including the operations we perform and structures we use).

Yet, math we know revolves around the abstraction of (discrete) language AND that we operate with things that we count. Even if we were slime molds (well within the same universe), we may have never developed the concept of integers. At the same time, there could have been 3D geometry without words.

Maths in other planets or universes can be different only if they have different sets of axioms and/or different logical inference rules.

Reader added context: humans are an inherent part of the universe.

We are a way for the universe to know itself.

Carl Sagan

"Through our eyes, the universe is perceiving itself. Through our ears, the universe is listening to its harmonies. We are the witnesses through which the universe becomes conscious of its glory, of its magnificence." - Alan Watts

The universe has a mind (human) and its current opinion is that math is inherent and discovered.

There's also the viewpoint that the universe is part of mathematics.

The mathematical model of the universe is part of mathematics.

But the universe itself might be made of mathematics, to the extent that it is made of anything at all.

it depends a lot on who you ask, so that part is human, and the part that you didn't specify is ambiguous enough that the answer can only be yeah no.

It can be both.

I've read other books by this author and felt that, for math books, they were overly political.

Is this one any different?

Yeah, I think the mathematical exposition was pretty good, but I couldn't bring myself to finish the book because of the culture war she kept trying to shove in.

Are you referring to this book in particular or her writing in general?

I am referring to her other book, The Joy of Abstraction.

Can you elaborate on what you mean by that?

Not the person you’re asking, but the reviews of previous books make similar complaints:

https://www.amazon.com/Joy-Abstraction-Exploration-Category-...

> This is not just a mathematics book, but unfortunately also a polemic of progressive politics. For example, on p.42 there is a helpful definition of “cisgender”, in case you didn’t know. Then, on p.319, the notion of isomorphism — ubiquitous in mathematics, and really quite simple — is explained by considering two categories induced by a partial ordering, one consisting of concepts involving rich, white and male, and another involving rich, white and cisgender. It is then shown that rich white cisgender women have the same status in the second category as rich white males do in the first. Just below this, there is a helpful diagram showing that each of these categories is a 3-fold product of the category consisting of two objects and one arrow, namely (people with structural power —> people who are structurally oppressed). Really, there are many simpler ways to get these ideas across.

Yeah, I try to avoid the culture war stuff, but this seems pretty unnecessary for a math book.

She used the same example in some lecture on youtube, too.

A good starting point for questions like this is the SEP article: https://plato.stanford.edu/entries/philosophy-mathematics/

Here's the irony.

In science and therefore reality as we know it nothing can be proven to be true. Things can only be falsified in science.

This occurs because if we make 10 million observations that verify a hypothesis we still haven't proven anything to be true because there always exists the possibility that a subsequent observation falsifies the entire hypothesis.

While we can't prove anything in reality is real, we can prove things to be true in mathematics. Proof is the domain of math and logic not science.

Therein lies the irony. We don't know if anything is real in science and therefore reality as we know it but we can verifiably know whether things are real in the universe of math.

It really puts the question in perspective: what does it even mean to be real?

But this just isn't the case, for your claim to be true we would have to be constrained to only do _inductive_ reasoning.

When we do _deductive_ reasoning, and we feel as though the reasoning has been done correctly, then we know something _assuredly_. Mathematics and empirical science are very different in this respect, that much of mathematics is purely deductive.

Inductive reasoning is by nature probabilistic so it can't prove anything.

See: https://en.m.wikipedia.org/wiki/Inductive_reasoning

The first paragraph mentions this. As science is statistical in nature inductive reasoning by being probabilistic suffers from the same problem.

Your claim was that we can only know about reality through inductive reasoning, and this is the part I'm disputing, not that science itself is not inductive.

Mathematical reasoning is deductive, and mathematics is part of reality.

> Your claim was that we can only know about reality through inductive reasoning, and this is the part I'm disputing, not that science itself is not inductive.

My claim still stands resolute in dispute of your claim. You can't know anything through probability/inductive reasoning. At best you can say something is "probably" true, but even this is a limit that's impossible to reach. If you observe something 1 billion times and that observation confirms your hypothesis, you never know if the next 10 trillion observations can deny your hypothesis completely. So even saying something is "probably true" can't even be done. Nothing can be truly known in science and therefore reality as we know it.

>Mathematical reasoning is deductive, and mathematics is part of reality.

It's only part of reality in the same way a fantasy novel is part of reality. It just so happens that it matches our observations. But observations are not always the same or consistent, how will you verify logic and math consistently hold true in reality? You would recursively use stats and science to determine it's veracity which suffers from the same issue as I stated above... you can't prove anything to be true with science.

Math and logic is an axiom of reality. We simply assume it to be true and there's no way to prove that it's true. But here's the kicker. EVEN if we assume science and logic to be true, we STILL can't prove anything to be true in reality. This is because of exactly what I'm talking about above... we can never know the true sample size of all possible observations... any amount of observations or samples we have is finite, but the universe is unknown and samples are potentially infinite. Therefore any sample could be 1/1000000000 of what's out there.

I'm not making any of this up. This is real stuff: https://en.wikipedia.org/wiki/Falsifiability

Quotation from the article:

"One of the questions in the scientific method is: how does one move from observations to scientific laws? This is the problem of induction. Suppose we want to put the hypothesis that all swans are white to the test. We come across a white swan. We cannot validly argue (or induce) from "here is a white swan" to "all swans are white"; doing so would require a logical fallacy such as, for example, affirming the consequent.[4]"

> what does it even mean to be real?

exactly the question which has to be answered before answer "is X real?"

I can already tell that a vast majority of the replies here are going to respond to the title and not the post

> The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real.

Solved!

If ideas are real, do you have any idea about entities that are not real? I mean, what is his exact definition of "unreal" things? Something that can not be formally described?

I'll let Parmenides take that one

> It needs must be that what can be spoken and thought is; for it is possible for it to be, and it is not possible for what is nothing to be.

This is not an alternative to people reading the article

Agreed. I just thought this little paragraph at the end of the article, which does directly address the title of the article, was amusing.

I've encountered this argument before, but also applied to other subject such as eg Latin:

> 3) math has an indirect usefulness which is a way of thinking that is transferable to a myriad of disciplines and solutions to problems in everyday life. And it is this third reason that makes math relevant for most people.

Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?

I’m skeptical of (3) as well, and yet I encourage students to study math through calculus, and to study Latin.

My feeling is that the utility of higher math, Latin, history and literature comes every minute of every day, as you experienced life as someone who has familiarity those things and your life will be richer and fuller.

This is decidedly not testable. And yet I still believe it.

There are plenty of ways to use this stuff in everyday life.

I took logic in college, and although the "logic as english statements" stuff was sort of confusing, the symbolic stuff like A&B = !A|!B stuff has helped with computers all my life.

It was only much later in life that I ran back into logic as english statements in a way that made sense as practical.

I read a book where they took apart the statement:

  If you loved me, you would take me to the movies.

Most people in relationships will respond to this with:

  Well, I just took you to the movies last week!  Why do you want to go again tonight, we had other plans!  etc...

But the book explained that with "If X, then Y" it was futile to address Y. You must address X:

  Wait, do you think I don't love you?  Of COURSE I love you!

...just hard to unpack this in the middle of an emotional situation unless you've studied the logic

I would highly advise against treating informal natural language statements as logical statements. It is impossible to know what someone who says "If you loved me, you would take me to the movies" actually means without far more context. The fact that it has the form of an X=>Y statement is at best a hint, but it definitely shouldn't be taken as literally as that.

I think you meant

  !(A&B)

Anyway… I made a video called “Why Think Mathematically?” as the first in a series called “Thinking Mathematically” on a YouTube channel years ago:

https://www.youtube.com/@thinkingmathematically

It might help to answer the question

An intro to formal logic class was the first thing that made me think of becoming a professional programmer. The class had a lab portion that used a program called Tarski's World that I remember as being a lot of fun.

The correct answer is

  Since when have you thought that I didn't love you?

Read The Gentle Art of Verbal Self-Defense for why.

I actually think point (3) is a good one, but am still extremely skeptical of teaching students more maths.

Basically, I believe that there's a heavy correlation between being good at maths and being good at solving every day (and not so every day) problems.

But I don't believe there's much of a correlation between being taught math at school for even more hours will make much of a difference. Most schools are terribly at teaching anything.

I feel I became much better at writing non-fiction after taking math at university (mainly calculus, linear algebra and statistics).

Going through proofs and proving things on your own really transferred to being able to better present arguments. The diversity of the math I learned has helped to reflect on things from different perspectives.

In sum this helped with everything from thinking more and better about the core issue at hand, writing argument chains in the correct order, cutting down on irrelevant stuff and more.

I've used this to significantly help the grades of both my SO and a family member, who both took non-math topics, by improving their hand-ins. I didn't know their field so was strictly improving the structure and presentation, and asking for clarifications where I felt the arguments didn't add up, and have them write down the answer.

I feel it still helps me a lot writing emails at work and similar.

That (3) is a very interesting issue. My suspicion is, it basically says "knowing logics is good for your well-being".

The skill of logical inference — even at the level of very basic syllogisms — is both very much underappreciated and underdeveloped in the American college population, at least from my personal experience. As good citizens, we all collectively should grab a couple of Martin Gardner's or Lewis Carroll's books off the shelf and give them a good read. I predict it will do much good... and if I'm wrong, it certainly won't do any harm!

Here is a study: https://journals.plos.org/plosone/article?id=10.1371/journal...

And, since LLMs are so bad at math currently, we may find that, by improving their math ability with gobs of synthetic data, we get improvements in general reasoning.

Thanks, this is a nice reference! Interesting that the abstract and introduction mention the lack of existing evidence - seems like an under-studied question?

One aspect that's quite easy to criticise about this study is that it uses existing groups of students with different levels of maths training. This means that there is possibly self-selection etc, and one may argue there might also be a causal effect in the opposite direction (e.g. folks that are good at reasoning like to do maths.)

> Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?

We know that if you try to train someone in math (or in Latin), and they do well, then they will also do well at other things in the rest of their life. Some people would like to give the credit for that good performance to the Latin training.

Brains model patterns in signals, math could be described as the brain's language for expressing higher order manipulations on all those models with communicable symbols. So it's kind of both real and invented, because brains invent, and brains are also just real objects and any models they form came from somewhere.

The statement that brains model patterns rests on the supposition that patterns to be modeled exist.

The notion of existence is doing a lot of the leg work. A epistemological solipsist can say things "exist" when they have a model of their subjective experience in which things can exist. It gets real funky because your interpretation of qualia results in a model that includes the brain and thus a model of how you are interpreting things as a model in your model.

If there isn't then I guess we're all just talking to ourselves anyways so it doesn't really matter.

Am I the only person who hates it when people say “anyways” when they mean “anyway”?

Perhaps it's acceptable in moderation, like "ain't". It's needed for a rhyme in a well-known song by Billy Joel, for what it's worth. On the other hand, there is a different song by Billy Joel in which "anyway" is used for a rhyme.

No, I'm not a Billy Joel expert or fanatic. I just happened to notice, all right?

That stray character doesn't change the meaning of the sentence, so yeah, stylistic complaints are subjective and beside the point.

We're not in school.

This introduction is a very pleasant read and adequately dense and easy at the same time.

... the typical working mathematician is a Platonist on weekdays and a formalist on Sundays

Philip Davis, Reuben Hersh, "The Mathematical Experience"

Meanwhile the actual working mathematicians, better known as engineers, still use geometry on weekdays.

Engineers are their own special species. They aren't just 'working mathematicians'.

No if you use sqrt(-1)

I think I heard that all numbers in the real world are irrational. So that means most of math is not real, except of course the irrationals like pi :)

I've come to the conclusion that all numbers in the real world are integers and real numbers are a human construct necessary evey time we select a unit that is too large. There is evidence (e.g. quantization) that at the most fundamental level, reality is discrete. Math is layer upon layer of abstract toolsets to operate on integers. In that sense for me, it is very real, but invented.

The fact that reality can be described by calculus would suggest that reality is continuous and not discrete. But I haven't seen any real evidence of either claim.

The more interesting question is are the numbers in the universe the subset of real numbers that are computable.

Why is that interesting?

Pi is no more of a human construct than any of the integers. Pi is inherent in Nature as are the integers, but more mysterious. If you accept the existence of the integers then infinity exists and therefore pi exists too; it does not need to be constructed.

There are no circular objects in fundamental physics.

Pi shows up in many physics equations, but that’s entirely due to our choice of units.

That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.

Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.

There are more irrational numbers, almost guaranteeing any number come across in nature is irrational

There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.

You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.

But pi is ONE "real" irrational number. The golden ratio is ONE more. That's TWO. QED the rational numbers 1 and 2 are "real"?

What's "real" about π? It's a tool. Circles are useful, but there aren't any.

It is better stated that: since there are "so many more" irrational numbers than rational ones, if you were to pick a real number "at random," the probability that it would be rational is zero. The "many more" and "random" ideas are made precise in measure theory (and elsewhere).

If you actually look at how real numbers are constructed. They are quite bizarre. The simple concept of the number line becomes a quite complicated set of sets that follow certain conditions.

(Sqrt(2) as a real number, is actually encoded as the set of all rationals less than sqrt(2) on the number line).

> The "many more" and "random" ideas are made precise in measure theory (and elsewhere).

Well... one of the consequences of that precision is the theorem that there is no such concept as choosing a real number "at random".

There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other? Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers? I would have thought that there being an infinite quantity of both, makes it impossible to compare the quantities.

There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa.

This is not the case for irrationals... therefore it is concluded that the infinity of irrationals is a larger infinity than the infinity of rationals.

See:

https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

https://mathworld.wolfram.com/CantorDiagonalMethod.html

There are also infinitely many rationals between any two distinct irrationals.

My favorite way of visualizing the difference uses the fact that every rational has a repeating decimal after some nth decimal place, and no irrational has a repeating decimal. Say you want to construct a number x, where 0 < x < 1, by drawing integers 0 through 9 randomly from a hat. Each integer drawn from the hat is placed at the end of the decimal; for example, if you draw 1,3,7,4 then the decimal becomes 0.1374. You then draw, say, 1, and it becomes 0.13741, and so on. If you could draw infinitely many times from the hat, what is the probability that you'll construct a number with a repeating sequence? That would give a rational number.

> There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other?

Mathematicians can even meaningfully compare infinities.

See eg https://www.cantorsparadise.com/this-may-seem-more-irrationa... or https://math.stackexchange.com/questions/474415/intuitive-ex...

You can also look at eg a uniform random variable on the interval between 0 to 1. The probability of hitting a rational number is 0%. The probability of hitting an irrational number is 100%.

> Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers?

No, that's not enough. There are also an infinitely many rational numbers between any two given irrational numbers.

> because there is an infinite quantity of irrational numbers between any two given rational numbers

Indeed, you've grasped the core of it. There's no rule you can write for irrational numbers such that "b is the next number after a", because there are infinitely many numbers between a and b that you'd be missing. You can't count them, i.e. you can't map them to integers.

Uncountable Infinities > Countable Infinities

Counter-intuitive as it may seem, math does have the notion of larger infinities.

This point of view is conflating two meanings of the world "real".

The "real" in "real numbers" has ultimately not much to do with our everydays notion of real. I'd rather treat it as an arbitrary name. You could as well call them "asdfasdf numbers" and they'd remain the same.

Last time I counted my kids, their number was definitely rational.

2 is also a real number.

Really not the best title for this book.

> Oftentimes, math is taught as a set of rules and processes divined from an authority in which learners must memorize the rules and strictly follow the processes in order to satisfactorily find the one and only correct answer

I was surprised later in life when the majority of people I talked to felt Math was this way (and a good thing). To them, math is a set of rules you learn to follow to the T and use them on other problems.

For me, everything a Math teacher conveyed was more of a recommendation, a suggested tool that I could incorporate into my tool box. The methods they employed to solve problems were a matter of preference to me, rather than rigid rules.

This way of thinking has always had some pros and cons. I never solved a problem the way a 'grader' was expecting. Some teachers loved the creativity and efficiency, other TA's just marked as zero. It made applying what I learned to other things, but on exams I would always be stressed with time because I would spend time on which way I was going to answer the problem.

That being said, I am very good at Math. scored well in HS/college, 165/170 Math GRE score, in a field where Math is important (Data Scientist).

If math is not real, it should be complex then.

Everything is real by definition

Is value real? Does it matter?

It is if you use pascal :D

Do you mean Pascal the person? :P

I was making a terrible joke about the old “real” type for floating point in pascal. Or possibly PASCAL :D

Unfortunately, it takes extraordinary effort by the math teachers to make Mathematics interesting to students. Even if some of the teachers are willing to put the effort in teaching this wonderful subject in a way that makes sense to students, it's not possible to turn all mathematics teachers into teachers who understand how mathematics MUST be taught.

I hope all these books, YouTube videos, and websites can help in making students curious about mathematics and explore further on their own. It is hopeless to even expect the school systems in US and a lot of other countries, to change the way their school systems work - teachers are not given enough time to spend on the topics in mathematics. It takes lot of extra effort by the students.

The actual reality is most students don’t have the cognitive horsepower for much past long division. I’ve seen adults struggle with basic algebra: “but what is X?” And so on. And that’s fine. Billions of people are going to live satisfying and fulfilling lives without ever knowing the chain rule.

Edit: I thought it was pretty advertent.

I think you inadvertently highlighted what I suspect is the real issue; a ruthless focus on the path to calculus to the detriment of other useful maths.

Other maths are more difficult to grasp than calculus. Calculus at least has physical applications and can be visualized.

I think I disagree! Calculus is great and all (truly), but logic, boolean algebra, probability, statistics, even linear algebra all have really interesting and useful insights and are very approachable. I'm sure there are other maths I've never encountered that can also be enjoyed with minimal pre-requisite knowledge, though that might be more in the 'abstract and difficult' territory.

The bits you need from calculus to approach those other subjects is also very minimal and approachable relative to the content of a calculus pre-req (where applicable).

Yes. Next question.

some of it is integer

You might as well ask "Are ideas real?" in the broader philosophic sense. Yes, they (with math as a subset) are real existents within consciousness. The question of mathematical ideas' connection to reality is epistemological. "Real reality" pertains to metaphysics - the externally perceivable stuff of reality that exists independent of our minds (as in, eternally prior to, and subsequent to, the existence of any humans or other beings with conceptual consciousness.) Math is an epistemological tool to abstract the causality of existence into a simplified structure amenable to consideration and manipulation by human consciousness.

A quadratic equation can be used to approximate the coordinates of the trajectory of a thrown object in a gravity field. I think the really interesting question is why that particular equation reflects that trajectory. Arithmetic operators - multiplication and addition in that case - in a particular order, are approximating the causal operations of existence that are actually at work. I think of this as the philosophy of mathematics and much more should be done to investigate it.

> Math is an epistemological tool to abstract the causality of existence into a simplified structure

The mathematics to accurately predict or relay reality is still complex enough that it’s often beyond us. You’re right in that it’s a tool to understand, but if we’re using simplified math for simplified reality, is it really epistemological?

As you suggest, math isn’t outside the boundary of philosophic investigation. It never was in the past, and I don’t think better approximations change that calculation.

You made me remember this article you might have read before: https://www.wired.com/story/our-machines-now-have-knowledge-...

Thanks, I’ll give it a read

> the externally perceivable stuff of reality that exists independent of our minds

How can any perception be independent of mind consciousness? And how can mathematics reflect anything other than that consciousness?

It's perceivable as long as a consciousness exists with sense organs. Existence doesn't depend on that perception. Existence existed long before any minds existed and will it exist long after. ("Long" is a euphemism here for eternally.)

> real existents within consciousness

"Within consciousness" doesn't belong here. Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist.

well… on the other hand: have been triangles ever experienced without a consciousness?

this "the idea of triangles would still exist" statement looks very reasonable, but can we ever proove it? you can not remove ALL consciousness to verify. at least a bit consciousness must be in the system to do the experiment.

yes, it very much coincides with all of our understanding of reality, that abstract ideas like triangluarity is independent from any consciousness. at least any consciousness known or conceivable to us so far.

so my extension to this statement would be - admittedly supernaturally-sounding - that: Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist, possibly because there is a consciousness above/beyond of the conceivable the universe to sustain their existence.

We're very quickly coming to the realization that information is a physical quantity, like mass or energy. (And can be measured and follows some sort of physical laws.)

If so, then consciousness doesn't need to exist for information structures to exist.

My view is that Math is real but the way it's represented is arbitrary. I'm sure that math could have looked very different than it does today and it would have worked better for a different kind of people. I find that some mathematical abstractions and ways of describing certain ideas are over-complicated and almost seem to be intended to confuse and obfuscate rather than to clearly explain. I feel similarly about certain fields of science; the ideas are really very simple but whenever you hear someone explaining them, the explanation seems to be full of gaps, rely on the audience to make certain assumptions, use custom terminology or terminology repurposed from different fields which can create confusion.

I often felt that being good at math requires having a natural talent for making the right assumptions to fill in any gaps or ambiguity in explanations. I feel like the language of math does not do justice to the complex, intricate ideas it tries to convey. On the other hand, being good at programming requires the opposite; it's about being able to resist making assumptions.

That's really interesting, can you give some examples?

I feel like almost all the descriptions of advanced (university level) math concepts are confusing when you first hear them. For example, the definition of a Taylor series according to Wikipedia:

"In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point."

This kind of description really throws me off. Especially "terms that are expressed in terms of the function's derivatives at a single point." first of all, superficially, the different meanings of the words 'term' and 'terms' throws me off and then even when I get over that, it's not clear what is meant 'in terms of' in what terms? What kind of relationship are we talking about here? I need to keep reading a lot more to fill that gap... In the meantime, I'm in a state of confusion and will need to re-read that sentence later to make sense of it once I have more info.

I'm not sure how to solve that problem to explain it more clearly but it feels like definitions should gradually build up without gaps. Is it even solvable? I would prefer not having read that definition at all TBH as it only serves to confuse me, it has way too many possible interpretations. I prefer to jump straight to the formula.

Completely agree.

I think there is a cultural component that has transferred throughout history whereby people need to signal their level of education to others.

You've probably had it happen yourself without realizing.

You hear someone explaining something in a simplistic way, and notice yourself wondering how deeply they understand the topic. Then when you are explaining the topic, you don't want people to question your own knowledge level like you did to the other person, so you use techniques to signal the depth of your knowledge.

This might be fancy words, or skipping over simplistic things.

And I think this just becomes second nature.

You can see it with programming languages. If I told you to rate a Rust dev vs JS dev, you are thinking Rust is harder to learn so they must be smarter.

It can also just be a challenge to imagine how you thought about a concept when you were initially learning it.

English is pretty terrible for explaining a lot of math too. Math is better understood visually, but back in the day you couldn't exactly share an interactive diagram.

That's some really bad wording indeed. But it may vary from one source to another. The more intriguing point of your comment is about assumptions. Can you provide a relevant example?

Mathematics is, in part, the study of universal traits:

(1) Ratios between things in the world.

(2) Logical relations between things in the world.

(3) Absolute distinctions between things in the world, in a nominative sense, which is to say in the sense that numbers can be assigned to things or elements of things.

Mathematics that relates to one of the three use-cases above is absolutely real, hence its unreasonable effectiveness in the natural sciences. (See Wigner: https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf )

When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal. A construct or game played with logic that has no prior or intrinsic relevance to the material world.

So when an application is found to a previously pure theory area (e.g. number theory and RSA) does math becone real?

Is realness not an instrinsic property but just a judgement on how useful something is?

Is being real a real property of something or just a construct?

Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.

The study of math in language is called semiotic, and thanks to a bunch of french poseurs it’s become a joke word among the genuinely intellectually curious. Nevertheless, attempts have been made to tie it all together, most notably by the brilliant and mostly forgotten Charles S. Peirce.

Whence the pragmatic maxim:

  Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

Pragmatically speaking, math is no more or less real than that.

> Using 'real' as the goal kind of misses the point

It misses the point of the question "is math real"?

Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.

Okay, here is precisely where the Simulation Hypothesis becomes useful and interesting. I know, I know. But bear with me for a minute...

The universe, as we know it, is simulatable. In other words, it can apparently be reduced to mathematics in precisely those three senses I outlined above. (If it helps, you can imagine electrons and other subatomic particles as bits of information in a coordinate space, which are constrained to operate in accordance with rules that govern logical relations between things.)

What mathematics is real? Anything that would relate to that universe or any of its constituent parts, in _any_ meaningful sense. We don't need to have found a use for it, and surely there's a great deal that is still undiscovered.

What mathematics is unreal? Whatever is, a priori, absolutely unnecessary to the existence of such a universe -- or, even worse, would break a universe described in terms of logic were it somehow made manifest.

Given that logics (of all sorts) are subsets of mathematics, I'm not sure one can convincingly hoist (any) logic above all other mathematics to make arguments.

> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.

Are you considering the set of natural numbers to be “unreal”? Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?

I'm not a finitist, but it's a highly defensible philosophical position.

As for this:

> Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?

Skolem solved that quite neatly already: Every "uncountable" set has a countable model. Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner. What I'm describing is necessarily an abstraction of an abstraction, though, so I'd consider it "unreal" by definition.

I know what “a model of a theory” (such as a model of a theory of sets, even one which asserts that some expression defines an uncountable set) means.

I’m not quite sure what a “model of a set” means. I suppose maybe you mean like, the set in some model, which is described by the given description of a set in the theory?

> Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner.

I don’t think this follows.

There are countable models of set theory. And for these models, there is (in the meta-theory) a bijection between the set representing the set of real numbers, and the set (in the meta theory) of natural numbers.

I don’t think this establishes that “in fact” there is a bijection between the real numbers and the natural numbers.

Rather, in any model of any of the usual set theories, there will be no bijection between the set of real numbers and the set of natural numbers. (Of course there will not be, because these theories entail that there is no such bijection.)

If I take a non-standard model of arithmetic, and for some non-standard natural number n, and consider the uniform distribution of (non-standard) natural numbers less than n, then for every standard natural number, the probability of getting that natural number from that distribution, will be equal. Are we therefore to conclude that there is a uniform probability distribution over all the standard natural numbers? By no means!

To be fair it's not too strange to consider both the natural numbers and the power sets to be to greater or lesser extents abstractions to allow us to reason about all the things we can logically put in one of them.

After all both of them are mostly full of things we will never need, encounter, or even be able to define.

I kind of think the core of the question of if math is real, is if abstractions are "real".

I was going to post something similar and then saw yours, and it was better. We can add that math is language, and sometimes language describes things that are real, and sometimes not.

> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.

I find that to be quite a bold statement. The question as to whether "mathematics is real" feels firmly rooted in philosophy. First, I would question what is meant by real and unreal. Second, I would question if the answer matters whether the existence of the real and the unreal are not yet discovered, or even imagined.

Your describing physics more than math in my opinion. But it's hard to say things like set theory isn't real when a lot of calculus works nicely and describes the world very accurately. That toolbox is heavily built on the "unreal" things in math your describing to be not intrinsically relevant. The weird parts of math often speak to some deeper thing your discounting. Hopefully I'm not reading into your comment to much

I think the tie between Mathematics and the real world is more of a fortunate coincidence than a necessary tie or even causality. Human will still study math (as we know it now) if it real world doesn't exist or it somehow becomes not describable by math, because there's the innate desire for logic and solving puzzles.

https://plato.stanford.edu/entries/social-ontology/

And yet irrational numbers are real.

“The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real.”

Alright great lol

As a data point, the author was interviewed on NPR's "The Indicator"[1], and the way this review summarizes it might not capture what her point was:

> CHENG: I'm not trying to answer whether math is real or not. I'm trying to show that considering the question at all leads us to interesting thoughts. And in the end, what I say is that, with all these questions, I don't think there are yes-or-no answers, and we shouldn't claim that there are. What we should do instead is say there is a sense in which - you know, what is the sense in which math is real, and what is the sense in which math isn't real?

> And the thing is, I think a lot of people who say math isn't real are using that to say, oh, so it's irrelevant and stupid. Why should we study it? It's made up. And what I want to say is that just because it is made up doesn't mean that it's irrelevant. And actually, the fact that it's kind of made up makes it really powerful because - well, it makes it really, in a way, more accessible because you don't need a lot of money to get it. All you need is an imagination. And I think that's a really amazing thing about it. And just like fiction isn't real, but fiction can give us insights about the world around us to highlight much more, specifically, things about society. And that's what I think is powerful about abstract math as well - because we're not constrained by reality.

I haven't read the book, but it seems like the title may be a throwaway question meant to pique the reader's interest and say "let's get philosophical about math".

---

[1] https://www.npr.org/transcripts/1193035114

This is an idea that goes back to Pythagoras and Plato.

Think of a triangle. Now draw that triangle on paper. If you look closely enough, you'll see "imperfections" in the triangle you just drew. Now ask yourself: "how do I know this thing I just drew is imperfect? Where did the idea of a perfect triangle come from?"

Plato would say the perfect triangle comes from the realm of "forms". This mystical place which is "more real" than "reality" because everything there is perfect and everything here is just a flawed approximation. Plato also said that this is the place where our souls go when we die, and we engage in "congress" with the forms and then return to earth, reincarnated. When we learn things, we aren't learning something new but actually recalling memories of the forms. This is why everyone knows what a perfect triangle is but no one has ever seen one in the physical world.

Because the Platonic soul (psuche) is different from consciousness. Actually, Philolaus (first Pythagorean to write a book) said that conscious feelings come from the combination of the mathematical soul with the body. So, Pythagorean soul can be viewed as the set of logical or conceptual forms— which constitute a person, yet can also be passed on from person to person. That’s a very different kind of reincarnation…!

What is “real”? Well, it’s just an idea… really

There is, almost certainly, an objective universe that exists outside of human minds, and it also seems likely that all of mathematics can be determined through objective mechanisms.

Shoulda put that on the preface just to save everybody some time!

Redefining terms until the question doesn’t matter is a totally valid way to solve a problem /s

Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. A rich enough language should be suitable for expressing both real and abstract statements. It is also interesting to think about mathematics from the point of view of programming language paradigms: is it functional (I think mostly), what is its type system, etc.

[1]: Galileo says that, roughly, the book of nature is written in the language of mathematics https://en.m.wikipedia.org/wiki/The_Assayer

[2]: A classical essay by Wigner on the "unreasonable effectiveness of mathematics" also brings up the language analogy. https://web.archive.org/web/20210212111540/http://www.dartmo...

I am no philosopher, and only an amateur mathematician…

But to me it seems that even when you consider math as just being language… you still run into the same philosophical problem when you consider what the semantics of that language are.

For example, you may believe that the practice of algebra is simply operations on stings of a language. But when you start asking what objects those sentences refer to, you end up asking what math is again.

This is where I finally landed with this question - there’s too much vagueness and importantly no agreed definitions of what is and is not “reality”

Math has no fundamental non-differentiable epistemology qua mathematics.

As you said there is no meaningfully discrete, bounded “object” that can be used as a universal reference - so the fact of relativity means that even if maths were “real”

I’m not even sure such a thing exists (who says the rules of the universe are uniform, or static as we’re only have a few centuries of poor observation) and if it does exist, if humans have the ability to identify it

This is a good point, but natural language has this problem to an even greater degree. What object does the word "consciousness" refer to? "Unicorn"? What makes a ship a ship? What if we include toy ships and drawn ships?

"What can math describe" is an easier question than "what is reality?" or "what is the nature of all concepts a human mind can conceive of or be interested in?". Not that it's an easy question either.

But we don’t expect natural language to be as precise and consistent as Maths. You could argue that maths was invented because natural language was too vague.

1. I do not think Galileo is either a characteristic example of a physicist or mathematician, or a particularly good example at all [1].

2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].

[1] For a criticism regarding galileo's mediocrity and historically unfounded idealisation https://intellectualmathematics.com/blog/the-case-against-ga...

[2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory.

[3] https://anthonybonato.com/2017/09/19/this-is-your-brain-on-m...

> And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].

I would not construe a small study of 30 people (15 math experts) as having the backing of the field of neuroscience.

The steel man position of mathematics-as-language would be something along the lines of "Mathematics is mostly a language + some other stuff". The paper in question only shows there is "some other stuff"---but that is also true of almost every non-contrived use of language. The paper shows that the other stuff for sentences about history and nature appear differently in the brain than the other stuff for sentences about math, but that is far from demonstrating that math on the whole is not mostly a linguistic phenomenon.

Personally, I think case studies of famous mathematicians or physicists discussing their thinking process (e.g. Einstein) is more convincing evidence against the hypothesis than fMRI studies.

They probably didn’t mention it because, like you pointed out in the sentence immediately after your question, it makes the philosophy ill-formed and dare I say uninteresting.

Are languages real? Sure, I guess. Why not. They’re made of concrete sounds and symbols that everyone (by design) can point to - they might even be the realest thing there is.

Are (math) concepts real?

As in like, if I rearranged all the matter that exists out there, would the value of root(2) stop being irrational?

Or maybe they don’t have to be physical to be real; maybe being “real” has more to do whether or not something has an effect on something else than its physical existence. Then, there’s a difference between how something exists and how real it is?

Etc.

Thinking of math as a language as opposed to a set of concepts isn’t interesting to the discussion of its reality IMO

> Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed.

Sorry, if this isn't clear to everyone, why does this result in the rejection of mathematics as a language rather than the rejection of the question of the realism of mathematics? The latter seems much easier with no downsides (that I can see), whereas the rejection of mathematics as a language has many downsides.

Of course, I do think of math as a (meta)language. Maybe my phrasing does not convey that well. So much for using natural language instead of math... ;)

I don’t think Einstein could have come up with general relativity if he had to rely on any other language than math. So I suspect math is not “just” another language.

So language isn't mathematics? Like physics "is" mathematics?

Are both language and math a third thing? Like maybe automata?

Mathematics is the language best suited to communicate Physics.

> The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

Translates to

> F=ma

Now translate that back to the original English without any context. I don't think your strawman is as succinct as you think it is.

Except the metaphor is quite misleading. Aristotle believed:

Force = velocity * resistance

considering that formal languages are part of math, there's little reason to believe natural language isn't. ft. LLMs.

Math is rational, therefore math is real.

Because ℚ ⊂ ℝ of course, but it is also not that far off from the conclusion of the book in that "math is real because it is an idea and ideas are real". In a non-mathematical sense, rational means "based on reason", and reason is the power of the mind.

I love your response.

But can math be both rational AND irrational?

In some cases he's rational, irrational, but not real: https://en.wikipedia.org/wiki/Math_fab_Mathonwy

Yes. But does irrationality imply non-reality?

I work with several individuals who are consistently irrational and frustratingly real.

Math can also be imaginary, but this also seems to be a feature of the universe.

Not for a numberphile.

Only if it's Schrödinger's math

This is a legit topos.

If anyone doubts whether math is 'real' or not - maybe watching this clip of these 4 men might clarify it (pay attention to the language they use to discuss the architecture of the universe):

https://youtu.be/AmUI2qf9uyo?si=6B6qA249jZqCVV5j

In my opinion, a more interesting question is "is anything other than math real?"

Is there anything that physically exists that is non-mathematical?

This is a curious statement. Mathematics is a description, not a prescription. When Newton says that the path of a projectile is parabolic, he does not mean that it is in the shape of a parabola. He means that a parabola is a very good way to approximate the shape of the motion.

The map is not the territory.

> Mathematics is a description, not a prescription.

But are you sure about this? If mathematics were not prescriptive then we should expect to see some evidence of that. So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics. Usually we would call such phenomena "supernatural."

It's possible in theory that nature doesn't have to obey mathematical laws. But if there's no evidence for it and if it doesn't explain any phenomena, then what is the argument for holding that particular belief?

Physics doesn’t obey mathematical laws. Mankind studied physics until we found mathematical formulas that describe nature reliably. There are countless math formulas that were discarded over the millennia after they were discovered to not adequately describe the physical phenomena they were intended to describe.

Math is a collection of abstractions. Some are useful in explaining natural phenomena. Some are not.

> So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics. Usually we would call such phenomena "supernatural."

You cannot contradict a description; it's not a "law" in the legal/prescriptive sense.

How would you even "contradict the equations of physics"? The real world exists and it's there; we use models to imperfectly describe (to varying degrees of success) its phenomena. In a sense, the real world contradicts those equations because they do not perfectly describe it; they are just an approximation.

> So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics.

That's kind of a tautology because physics is rules for what we observe. If we observe something contradicting the rules then we come up with new rules to describe the new thing.

The most straightforward evidence that mathematics is descriptive and not prescriptive is Einstein's General Relativity which illuminated that Newtonian mechanics was not prescriptive. Similarly General Relativity does not prescribe what happens at quantum scales.

Well, what is matter made of?

That's a billion dollar question. If you have the right answer, there is a good chance you'd win a Nobel Prize.

Well, what do you mean when you say something that "exists physically" is "mathematical"? Depending on your answer, the answer to your question could very well be "everything".

For example, we could ask whether anything that we think of as "existing" would be missed by simulating the equations of physics with the empirical starting conditions of our universe.

There are real limits in what humans know about the universe. But it's possible in theory that everything that is "real" follows from some differential equations.

In theory, maybe. I don't think our models of the universe are anything other than imperfect but useful approximations though.

Could your question be rephrased as "take the real world, with its things we don't have an accurate model of (which is most of them). Would we notice if we replaced this universe by a tidier one, perfectly described by a math model?".

It's an interesting, scifi question alright. I don't know if it has an answer.

> Is there anything that physically exists that is non-mathematical?

That's basically asking if there's true randomness.

What would it mean in this context to be "non-mathemtical"?

I mean that it can't be completely described by mathematics

Can reality be completely described by mathematics, or is every model just an imperfect approximation?

Well, think the other way around: if reality didn't keep the relationship between ratios as a coherent structure, it would be a soup of pure timeless randomness, producing lots of short lived Boltzmann's brains... so, we wouldn't notice.

I remember reading of a theory that kind of said that in any large enough random set, over long periods of time, you can identify clusters which will have consistent and predictable behaviors, but only within the bounds of the cluster and for a set period of time, eventually drifting to something else.

I thought there was mathematical backing of this observation.

Which means it could all be just random, but we happen to exist at such a small scale within it all, and for such a short period, that we're able to find laws applicable to our pocket and our time period that appear to be true and can predict behavior.

Thought it was interesting.

Definitely interesting. I dislike thinking about such things though. I've been stuck on this one for a while:

Descartes posited, "I think, therefore I am" as a reassurance of our existence in a given moment. While this anchors our immediate awareness in some semblance of reality, it doesn't necessarily offer solace regarding the past or the future.

Imagine, if you will, an expansive universe characterized by its sheer randomness and chaos, existing for eons. If, in such a universe, a consciousness can spontaneously emerge and dissolve—much like a fleeting set of physics—how can we be certain of its duration? Does it persist for a lifetime, or does it flicker for just a moment?

Our memories might suggest a continuous existence, but what if they are merely ripples in this vast sea of randomness? It's an unsettling thought: could it be that we exist only in the fleeting now, borne from the precise alignment of countless variables just to perceive this very instant? And that perhaps, a moment ago or a moment hence, we simply weren't and won't be?

While such thoughts might be disconcerting, their implications are, in a way, irrelevant. Even if I could unequivocally prove this transient nature of existence, our ephemeral consciousness would barely have time to grasp it. Our understanding would be obliterated, only to be replaced by another random state.

And yet, if this fleeting moment is all we have, I'm honoured that you spent it reading this randomly generated comment. Not that you had much choice in the matter.

I think it was Shakespeare who said: it was the best of times, it was the blurst of times.

Based on the Amazon reviews, this doesn’t look like a very good book.

“A Lot of Non-math Digressions”

“Way too wordy and discursive. Boring format.”

“It's hard to imagine that someone would write a book ostensibly about math that has so much personal, political, social, and just plain silly off-subject commentary. This book is so interspersed with that kind of commentary that it is difficult to focus on the math.”

“The book has some helpful insights regarding teaching elementary students basic mathematical operations. However, it is filled with so many abrupt political digressions, often incoherent, and ironically, occasionally illogical, that the entire work is barely readable.”

Math is integer, unless declared real.

Definitely found the Fortran guy.

Double down on C.

That's irrational and imaginary. Oh, and the cardinals will win the series.

How does the word "real" serve practical purpose, besides for the fun of logic / language game? It's a tool that we use and we found useful.

No. Math is Complex.

No. Math is Surreal.

Elephant to the melting clockth power equals what?

It's a tool. So it's as real as any other tool we have. It helps us bring about our thoughts in to the real world. We can't touch it, in the same way we can't touch language, but it's useful and that makes it real.