What Is Math?
smithsonianmag.com
smithsonianmag.com
Also if you enjoy thinking about this topic I recommend Neal Stephenson's book Anathem, which I just finished. It is a complete slog at the start, with endless boring architectural descriptions. I listened to it in audiobook form which made that section much more pleasant.
I'm with Max, from Pi. To restate his first assumption: Mathematics is the language of nature.
Cue Clint Mansell...
Consider an EM field. Do you think there is some daemon grinding out complex exponentials as it propagates? In reality some spinny thing is rotating and moving through space. What those terms actually are isn't precisely known but the phenomenon closely fits a mathematical model we can elaborate on to do useful things. That doesn't mean it's how reality works.
And that's before getting into categorization issues. Does Pluto "exist" as a planet? It'd be pretty non-sensical to say it stopped existing when scientific authorities changed its designation. But then was Pluto really a planet before that happened? Or was its planetary status a figment of collective imagination?
I agree categorization is a (almost?) purely human concept. I'm not sure how that relates to my question.
Why does this relate to your question? Because we cannot prove _anything_ not to be an artifact of our perception, nothing that we conceptualize is necessarily inherent in reality. In a strict sense, reality is unknowable to us. In a practical sense, none of this matters because math seems to work well. Still, if we wish to be undogmatic, then we should approach questions of existence or non-existence with humility and not base ontology entirely on perceptual illusions.
Edit: To clarify better, the way you solve math is that first you solve the problem, then you figure out how to communicate the solution. This can either use language people created before or you create new language, you can just use English or you can just draw pictures and have no words at all. Either way you are doing maths.
A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).
Language itself is an exclusive operation on the possible coordinates of anything within a universe.
E.g. the word “carbohydrate”, or it’s equivalent in different languages, describes a concept; but the concept itself exists independently of any language construct that describes it. Similarly, in math, a concept e.g. “prime number” is independent of the many ways and many languages you can use to describe it. It just is.
This is very interesting. What is language? When does language evolve? Is language itself consciousness?
Whenever there’s a leap in language is it because something had to be constructed to help assign an abstraction to a coordinate in space where thoughts exist?
It appears we’re languages all the way down.
Or analogicalmachines? Lol.
Lots to unpack here - some with what many would call not nice Consequences.
What happens when a minority elite (unsure how else to describe it?) disciplines require a language that the majority lack the finesse to use? Example: math used in physics, or programming in computation. If the languages being used introduce more complexity without the ability to reduce net complexity then it damages the communication of the specie using it. I’ve been thinking about this quite a bit on our trajectory as a species.
Complexity in language leads to information silos which leads to various existential risks based on information and miscommunication.
Also interesting coercion / authoritarian emergence comes out of those creating a new language to reason about an abstraction which leads to a success that negatively impacts those which do not know the language and if the space to understand said language is sufficiently far enough you have much room for manipulation. Also, who gets to say which language is the right one? What happens when a new language is created that is more advanced and the majority disallow it preventing an evolution? Interesting dynamics.
In this analysis, the "explain quantum phenomena" game is just a language where the rules are especially strange and prone to pitfalls. Deploying vocabulary from mathematics, a system with very well-defined and mutually-understood rules, enables us to navigate this space.
I’ve thought about what you said recently and it comes down to “well-defined” being relative. Usually to those that are defining it. It’s not objective. Additionally that’s enforcing or coercing a game unto others that they may not wish to engage with or may not be able to conceptualize it.
Some people prefer characteristics over others. Have predispositions to likings and not likings.
Using a language is in of itself agreeing to a game.
If a thing requires consent of will to proceed as a rule, it’s a game.
Language, or any system of rules would be part of agreeing to a confined space of possibilities and “game-space”
Interesting to ponder nonetheless at a different abstraction. I keep trying to go “deeper” behind the language itself, I arrive at signal and then we’re in a puddle of discrete and non discrete values. Fun.
Turns out dad was right.
I say alternative because it shifts the focus away from person-to-person interaction and more towards impersonal social processes that construct what we think of as "objectivity". That's not to say what you're referring to isn't also an important part of the picture so to speak.
On a serious note even if objective mathematical entities exist, which I personally don't believe is verifiable, it'd be really something to see how they can maintain said existence while satisfying Gödel's proven theorems.
Why?
Finding out that Pi is irrational and then transcendental are discoveries. I won't be able to make anything work if I fork mathematics to make it rational.
Same thing that finding out that if you tie a string to a stake and then go around keeping it tight, you have to walk a distance that is two times pi times the length of the string to get where you started. I don't need a language for that, just more pieces of string.
Pi is not calculated by running around in a circle with a string (nor by any other real-world measurement), it's calculated by setting up a rule-based system (math) and following the rules to derive its value. You can get pretty far in the real world by using a rational value for pi - ten decimal places should suffice for atomic-level precision, I believe.
One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché.
So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant.
Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.
This is entirely oxymoronic and double thinking and exactly the type of mental gymnastics that manipulates what truth is. Physics is based on the physical because it was built upon observation of reality using our senses. Yet modern physics is based on the imaginative and has nothing to do with the physical.
Any measured unit is imaginary. The only thing that exists is the measure.
Exists is based on physical. To be is to exist. To exist must be within physical reality. Without this distinction then we might as well give into the esoteric notion that everything is entirely a mental construct and nothing actually “is.”
How far we’ve come from the physical to the mental and then lack the nuance to distinguish as such and furthermore blur the lines is a disgrace to critical thinking and cognition.
This is only a general comment because I don't understand anything you've written so I can't address your points, which undoubtedly must exist.
It is this complete logical decoupling from any physical, observable reality, that I cannot accept mathematics as anything other than an, ideally, internally consistent, logic used to define and discuss symbolic logical objects. And I am of the opinion that logics are merely invented by conscious beings as means to systematize areas of discourse and to derive internally consistent inferences and implications built there on.
I would agree that if irrational numbers were prohibited from being inferred in the logical system of math then some things would not work within the logic. Because a change in the axiomatic or inferential rules of a logic are changed the logic as a whole changes. But, if, as you describe it, math was forked to be limited to rationals, and the precision of those radicals was calibrated to be on the order of 1/10 of a Planck length, I am not convinced that any equation attempting to describe 3 dimensional space would be inaccurate. However, I could wrong.
It’s similar in my perspective to a nuance that exists pervasively today within scientific models. Just because the model allows a prediction doesn’t mean that the model reflects “base reality.”
Our models, our tools, are the results of our observations. If it works to provide a prediction, then it becomes useful. However I feel there’s a disconnect as the nuance of a model vs what is, has been lost among many.
The map is not the territory.
It’s very interesting to see science use mathematical models as objective reality and orthodoxy. I’d say that isn’t science, but words change in meaning with the zeitgeist. Unfortunate in many ways.
I like the way you framed the argument.
Any recent things you’ve read in this realm of subjects that were interesting to you? Any favorite past time papers/books you’d recommend?
For some topic adjacent past time papers, a lot of my comments concerning logic come from the research for my programming language, so I have been immersed in logic and proof theory work for the last six months pretty hard. I think any of the course note PDFs from Frank Pfenning (CS prof at Carnegie Mellon Univ.) are great read in general (and are easily found by googling Pfenning logic course notes and just looking around). If you like video lectures, or just listening to them, any of Pfenning’s lecture sets from the OPLSS session, which are all on YouTube are wonderful, particularly the 5 lectures from 2017 on Substructural Type Systems and Concurrent Programming [2] Also, Noam Zeilberger’s OPLSS lectures on Refinement Types [3] are great and his dissertation was a great read [4]. Finally, Neel Krishnaswami and Dunfield‘s 2 papers on Higher Rank Bidirectional Type Checking [5] was really good (and like all the above the cite lists are a trove of good stuff).
I literally could keep going for hours on great reads in logics and type theory, so if you want more in some area I’ll provide.
Cites: 1 - https://plato.stanford.edu/entries/philosophy-mathematics/
2 - https://m.youtube.com/watch?v=ZW-VWWa1o9g
3 - https://www.youtube.com/playlist?list=PLiHLLF-foEeyS8V4lwN1q... 3a (course notes) - http://noamz.org/oplss16/refinements-notes.pdf
4 - https://www.cs.cmu.edu/~rwh/theses/zeilberger.pdf
5a - https://arxiv.org/abs/1306.6032 5b - https://arxiv.org/abs/1601.05106 5(2019 Version) - https://www.cl.cam.ac.uk/~nk480/gadt.pdf
I think it’s important to expand further in this direction. Confirmation bias or not, I’m compelled to dig further under these ideas. Thanks.
I notice you’re on a throwaway. If you’re on any other platform I’d enjoy staying in touch. My profile has sigs.
Not very divorced. I can glance at my coffee cup and notice that there is a distance around the circular edge and another if you were to go straight across. Maybe they are just having a trial separation?
But it is uncommon in my experience to actually define Pi as this ratio, if you build math axiomatically; Rather, the "shortest way" to get to pi which is well defined is by first defining the exponential function ( \exp x = \sum_i=0^\inf \frac{x^i}{\fact i} ) with all the pre-requisite for that (numbers with order, addition and multiplication; limits and convergence; then imaginary numbers). Then you define pi to be the smallest positive number such that exp(2pii) is 1, and e to be exp(1). All the properties of pi follow "easily", including it being the ratio of am euclidean circle's circumference to a diameter.
The thing is, in math, all of these things end up the same regardless of where you start; Whether you start with a geometric definition of a circle and work hard to discover said ratio, or you start with exp(), you'll end up with pi=3.14159.... and it having the other properties. It is in that sense, not arbitrary.
You could (and would) take a step back, and say that being euclidean is arbitrary - which is true; but any description compatible with euclidean axioms will get the same value of pi as the ratio of circumference to diameter; and that value will be the same of the exp() value that has no concept whatsoever of euclidean space.
It is in that sense that math is "discovered" - there is no euclidean construction in which pi is different. There is no peano-compatible (set, games, p-adic, or other) construction of natural numbers in which primes do not exist. The peano axioms themselves are, indeed, logically arbitrary. But the "discovery" is that "peano -> existance of prime numbers" -- and that does not depend on language.
If following the logical inference chain that leads to determining the geometric concept of a circle and then inferring the ratio between the circumference and radius is a discovered thing, then any and all logical inferences which can be proved are then discoveries, but I don’t think that argument can be taken seriously.
Given we had no language and communicated telepathically I’ve imagined that:
1. Using a feeling
2. Using a mental image
May be enough to form abstractions but that may in of itself be within the set of languages.
Said another way though, does language require symbols? It’s certainly possible to imagine language without words and we live in a time where symbols are abstractions for entire concepts within the noosphere.
Language is fundamentally based on abstractions.
Abstractions are fundamentally based on mind.
If the mind conjures abstractions then is it fundamentally a language no matter what? Probably.
What are the ways that we may use language to communicate? Far more than we think of today. Every time we extend the abstraction we are essentially evolving as a species. Almost to the point where our evolution is entirely dependent on our ability to abstract novelty continuously.
https://en.wikipedia.org/wiki/Aphantasia
There are many who are unable to form images in their mind. Aphantasia isn't rare. But it's also not universal and people with aphantasia somehow manage to communicate to those without well enough most of the time. Perhaps then language isn't just visual images or purely mathematical abstractions.
I think the view of Wilfrid Sellars, basically language as a collectively-authored set of referents to a non-visual conceptual "picture" of reality, matches well with what you're saying here.
Interesting stuff. Thanks.
It's clear that every animal has the abstraction "food", which applies to groups of molecules and atoms, and does not require anything like a language.
How does one find ethical or religious problems and questions in pure logic? Logic is fundamental and proven to be able to apply in reality. Ethics and religion operate over multiple levels of qualitative transitions and cannot(?) be fundamental.
Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)
Because DoJC exists outside of space and time in minds of believers? It just doesn't have a physical counterpart and does barely have any logic to evolve/deduct from it, for us to predict events based on DoJC as if it was applicable to parts of reality. It is understandable how someone could ask "stupid" (not really) questions about maths, but when philosophers start to draw parallels, it is easy to lose track of what they are even talking about. Can someone explain what they meant exactly?
The Observer Problem is a good example. I experience life from this singular human's perspective. Why this specific one? What mechanism dictates which human my "observer" occupies? Why even a human and not an animal?
The word "soul" is a poor descriptor because it is associated with so much more, but the Observer Problem is real - or at least it might be.
I heard a theory is that there is another higher order time dimension, and all of our "observers" are actually the same one living in some sequence "simultaneously". ...but that doesnt really solve the dilemma either because even that sequence would need some mysterious ordering mechanism.
...but in all that there are core questions about what is time, what is space, what is energy, what happens to the universe... etc
One thing I tell me daughters is that death isn't scary. ...because we exist inna limited space and time. We should be no sadder to not exist in the future than not existing 10m to the left. We cannot be eternal, but if you recognize that time is just a dimension, then our existence is permanant in the history of the universe. Nothing can undo what we did and who we were.
Sorry for off-topic, but The Observer is my great interest as well. Why not an animal? Sure, why not. Imagine that your ears, eyes, taste, etc are shut down and your consciousness too. Now you feel yourself as a rock. Why not a star? Why not a computer game character? There is a bunch of problems to think of before you answer "no". The same for "yes", because there is a theory that it's a human mind that thinks of that Observer concept and makes itself "feel" it, but... it is not real, a perception mistake required for a mind to function in reality, like many others.
This has something to do with models, but it is hard topic on its own. It seems that The Observer is one of things that don't have a quick answer and may not have one at all.
I stopped at this crossroad:
- distinct and only me (1)
- distinct and multiple (N)
- a field-like phenomenon (~inf)
- a perception mistake (0)
Is it? There's no way to demonstrate this.
That's rather a simple one, every point of experience is "this specific one" for itself, but in reality there are no specific points. If you were a fish in previous life you would only know that and the question would be the same.
If you mean "why" in terms of "how", the mechanics of it, well as far as I can see it's way bigger than our intellect and in turn language can handle. One assumption I have is that at the moment of death we will get it, not intellectually because the brain won't work but rather experientially.
>our existence is permanant in the history of the universe
Tell me how do you really, existentially separate yourself from the universe?
Exactly, but why am I not a fish?
Nothing is fundamental in logic, 2500 years later and we still haven't "figure out" Zeno's paradoxes, and as such we haven't been able to refute either Parmenides's nor Heraclitus's way of looking at things/reality.
Yes, the statements brought up by Aristotle a little later allowed us to be pretty efficient further down the road (without Aristotle's "Logic" we probably wouldn't have had "science" as we know it), but there's nothing fundamental about them. And I suspect Aristotle himself knew that, almost the only way he could attack Heraclitus's views were "ad hominem"s, there was nothing "fundamental" in his explanations of why Heraclitus was wrong and why he was right.
Maths is about creating an abstract framework with the mind (which somehow exists without typical bounds of time and space like everything else we can touch). Then you use the framework to sometimes solve real problems.
Logic only exists in our minds as well as other things mentioned. The problem which arises with making logic fundamental is that it does not do much for things which matter most in our lives. Like logic can not bring you happiness and joy. I can't make your experience sweet or ecstatic. Nothing wrong with logic it's just limited. Once people see those limits clearly they start exploring life beyond logic.
In either case it's hard for me to see what Platonism can tell us about the world or ourselves given all we know about math and it's many gaps.
As to the epistemological quandary, I always return to Geometry or Visual Mathematics. Consider the infinite ways of calculating the digits of PI. In "Playing Pool with PI", Galperin demonstrated an analog computer that could calculate the digits of PI using only the perfectly elastic collisions of two billiard balls on a pool table. It works for all integral base number systems, and can even be extended to irrational numbers. And recently Google Brain's Adam Brown made an equally stunning observation: Galperin's bouncing billiard balls look identical to the Grover Quantum Search algorithm!
Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search
Fast-forward ten years. I am a self-taught programmer and now find mathematics absolutely fascinating and beautiful, and I see its applications. So I'm taking math classes on Udemy, buying hundreds of dollars of textbooks, and trying to teach myself.
I can't help feeling like I got a little robbed. I understand that teaching is hard, and some kids just don't care to learn certain things and it's best for them and society to force some absolute fundamentals down their throats. But at a certain point around middle or high school, I think there needs to be a hard pivot to focusing on motivating students. If you do that well, they will have the ability to teach themselves and learn for the rest of their lives. If you don't they either end up like me, playing catch up a decade too late, or they just leave entire fields of knowledge untapped forever. With some subjects that's inevitable and probably ok, but math doesn't seem like it should be one of those subjects.
1. A clear explanation of a symbol system and how math is a symbol system to deal with issues of quantity, shape, structure, and logic.
2. Frequently converting between diagrams/drawings and symbols (show "x x x" at times and "3" at other times). I actually have a concept for a game I want to program that does this.
3. Showing multiple concrete applications of different ideas, and lots of word problems. In retrospect I think this was always reversed in the American system. We would learn the symbolic and algorithmic aspects of a math concept, then use it to solve some word problems. Once I went back and started teaching myself some of this stuff, I read that the Russian system is quite the opposite: lots of word problems, multi-faceted problems. I suspect I would have learned from that style a lot better.
EDIT: One more point I should make. I scored in the 85th percentile on my SAT math. It is probably not a great sign for our system of education that I could do so well with nothing but the most rudimentary understanding of what I was doing.
Euclid II also develops most of Algebra geometrically, so there's much conversion between shape and symbol baked it. His method of computing the product of binomials is far superior to what's commonly taught.
So there's a good tradition of the thing you're looking for!
What you want is methodology like "just in time learning" or "problem based learning", which is very powerful but also very slow. You have to greatly reduce the content you can cover in a class. It's good for some students, boring for the smart ones (who solve your fancy word problems instantly), and the worst ones don't pay attention or do homework either way.
Asian education systems like India, China, Japan are very rigid and based on rote practice and memorization. "Prep school" there means evening schools where they just go do yet more memorization and rote practice to help pass entrance exams. Yet they produce many brilliant mathematicians too. Of course the students are motivated by parents and society, not love of cramming.
The ideal would be teaching that is individualized and does the best it can to get through to each student (there are plenty of efforts and even "edTech" that tries to do this, especially at lower grades). Without that it's always a trade-off in which students get the most value for the time spent.
I was "good at math" in high school, to the extent that I could crunch through any problem in the math or physics textbooks. School math existed mainly to serve the science and engineering students. Now I was happy to use math, for instance in my electronics and programming hobbies, but that wasn't math as an end unto itself.
But proofs are what really made math come alive for me. I don't think this makes me a freak. A lot of people I know from my generation said that their favorite high school math course was geometry, which was heavily proof oriented at the time.
Sadly, the contemporary K-12 math curriculum is sorely lacking in proofs.
Now, what about the students who would be more motivated by usefulness than by abstract theory? For those students, I suggest looking at how people in the so called "real world" actually do math, and work backwards from there. My days of abstract math are behind me, but I am still one of the "math people" at my workplace, in an industrial R&D department. When I do math, I'm never far from a computer. Yet school math is still taught by mainly pencil-and-paper methods, even if those have been translated to online forms.
For many of my math tasks, I start by playing with numbers, e.g., computing a function and graphing it in a Jupyter notebook. It brings tears to my eyes that K-12 students are not exposed to this. For one thing, it's fun, and it's an honest portrayal of how people actually work. For another, you can brute force your way through a problem even if you've forgotten some particular formula, making it more likely for someone to use it later on. Graphing calculators are of course a thing, but they hardly go far enough.
I think you're onto something, because teaching myself programming was precisely my backdoor into math. I remember a line in the Clause Shannon biography "A Mind at Play" that said he was essentially the same way: he wanted to apply his mathematics and understand the proofs/structure behind it, but essentially (as the title suggests) it was a form of play.
Sorry, but following this logic we will end up with a conclusion that almost no school knowledge is "useful". How useful is playing volleyball? Reading poetry? Painting? Studying history? Biology would be reduced to first aid and basic medicine. I have those discussions with my daughters who learn how to do long multiplication. How is it useful if even I who know how to do it would rather use a calculator on my mobile instead? We would reduce math to basic operations on a calculators, probably nothing more would be considered useful by the majority of the population. The same apply to all other subjects.
The concepts we are taught today, and the way we are taught them, would be foreign to Pythagoras. According to wiki, symbolic algebra only emerged in 1200+ with the Arabs (which probably links to why we don't use Greek by default for algebra but we do with geometry).
[0]] https://en.wikipedia.org/wiki/History_of_mathematical_notati...
There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.
It's also worth noting that physical and mathematical existence are based on completely different criteria. For non-constructionists mathematical objects exist once they are not demonstrably contradictory (although the absence of contradictions often cannot be proved in an absolute sense). In contrast to this, for physicists an object exists once it can be measured, where measurement is ultimately tied to sensual experience. There are also theoretical entities in physics that cannot be directly measured, but their existence is usually downplayed, they're not supposed to "really" exist but only as theory-dependent entities. In any case,the two "kinds of existence" are very different from each other.
You cannot show me threeness. You can show three things and tell me to generalise, and if you keep generalising you'll end up with a consistent-ish framework of sorts for your observations that can be applied to certain other physical experiences.
But fundamentally this is an exploration of the consequences of psychological processes - like perceptual grouping, and inductive relationship inference - not observations of external phenomena.
It doesn't seem that way, but there is no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition.
For a long time, up until recently, mathematics was way ahead of the applications of mathematics to physics. In Ancient Greece there was a general consensus that infinity and real numbers do not exist. But then some people found out that the side of a triangle must sometimes be a real number. However, you cannot ever measure SQRT(2) precisely. Whatever number you extract from the physical world is only a rough finite approximation. You need to represent the number in a different way and solve the problem algebraically. Many scientists in Ancient Greece rejected this idea vigorously. Still, real numbers are very useful for describing the physical world, so useful that we couldn't possibly do without them today.
Imaginary numbers are another example. They were ridiculed as abstract nonsense when they were described for the first time and widely conceived to have no physical reality or application at all. Despite all that, they play a vital role in modern physics.
There are many more examples like that. To cut a long story short, at least until recently mathematics was always ahead of physics (now they seem to go more in tandem). This fact makes the idea very implausible that mathematical structures are merely useful abstractions from the physical world we invented to describe it. It simply doesn't describe what happened in mathematics. And I find the idea that mathematicians just came up with arbitrary imaginations equally implausible.
> no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition
That is true for everything, it's just a radical skeptic position. Nevertheless, mathematics has the highest standards of rigor for proofs among all disciplines.
That which is measured, exists.
No measurement itself, as in of itself as a thing, is a thing that exists. You can measure 3cm but “3” and “cm” don’t exist. Virtual values assigned by consensus. All axioms are agreements. Truth or existence itself is convergent and resembles certainty only in the majority agreeing to it.
Imagination allows nothingness (imaginary things) to be measured.
Imagine any unit, give it a “rule” and now it can be measured!
Pretty interesting bridge between the “imaginary” and the “real.”
There’s so much nuance on what existence is, and isn’t, or even not is nor isn’t is depending on if you subscribe to classical logic based on what we currently describe as mostly Aristolian or if you use many-valued logics which are making a come back yet rooted in the Vedas...
I find this extremely profound and it’s one of my top focal areas of study right now.
It’s also related to my observation of disagreement, chaos, and behavior of the human animal. It seems the zeitgeist of the egregore lacks the ability to adjust perspective and see truth depending on the “rule/roles being played” (all disciplines are games/acting in a way).
Tons to unpack.
(What is our substrate? Turtles, all the way down.)
Institutions these days never seem to pass up a teachable moment when it comes to reinforcing the precepts of dialectical materialism. e.g. "If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)" It's so tedious as to make me doubt their humanity, but I'm glad this girl sparked a popular discussion.
Or, more on-topic from the same: https://vimeo.com/77451201
I especially enjoyed the replies by Eugenia Cheng [1] on points 3 and 7 (from a link in the submitted article).
[1] http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-tw...
I did find a couple of things quite refreshing in the article, though: (1) Wigner's essay is delightful and I'm glad it's seeing a wider audience. We should be reminded of it every time this conversation comes up, if only for its last paragraph. (2) The reference to the fact that Galileo proved that falling objects accelerate at the same rate, not by some experimental demonstration, but wholly within his mind---and it's a proof that any middle-schooler has no difficulty completely understanding. As a reminder that mathematics isn't symbols and rules, but a way of looking at problems (Eugenia Cheng's book, How to Bake Pi, is an excellent look at this way of thinking and approaching the world.). (3) It managed not to get bogged down in the whole pointless and hypocritical "what is math good for" instrumentality debate, that these sorts of questions tend to devolve into.
Up to high school they taught it to me by textbook, mainly teachers who didn't know or care about it. I found beauty in it and loved it despite them. I suspect if we taught it less relying on authority and grades, more engaging and based on curiosity, we'd appreciate it more. Instead of asking what it is, we'd do it.
Even people who don't play music don't ask what it is, like it doesn't need to exist. They probably would if we just taught music theory from textbooks but not singing and dancing.
Yes, those are all meaningful, interesting questions that people have pondered for a very long time.
Math is a synthetic a priori judgment, a transcendental object.
I can only believe that this fashion of thought is from the unwarranted development of the nihilist project which came nearly 100 years later.
Bertrand Russell wrote in his historical treatment of philosophy that much of philosophy is forgotten and rediscovered throughout history. I'm skeptical of the argument but less so everyday.
>Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist...
Roger Penrose states in his book "The Road To Reality" that he is a light Platonist.
If only.
A more mature comment would be to point out an instance where there is some philosophical challenge to transcendental idealism that came from 'new mathematics'. Though I find the possibility of this to be impossible. I would expect at least the German-Idealist Godel to write about it.
If you're talking about Gauss's non-Euclidean geometry, that's completely compatible with transcendental idealism, in fact Gauss took his non-euclidean geometry from the second chapter of The Critique of Pure Reason. He read Kant's passage on the triangle and the line in respect to the sum of angles obsessively.
On pure existence proofs, mathematicians have generally agreed that something can be proven to exist by proof by contradiction on an infinite set. The result is that we say that it exists even though there is no way to find it, and no way to verify that you have found it if you are presented with it. Nothing in Kant's philosophy had anything useful to say about the debate about whether to accept this kind of proof.
None of this contradicts the synthetic apriori of mathematics, in fact Kant's critique strictly tells you that analysis IS possible and guarantees its legitimacy. I would look to Wittgenstein and Godel for that expounded application of the transcendental idealism.
The Pyrrhonists, Diogenes, Descartes and others famously believed in this; but Kant fixes this, making the discipline of philosophy possible by proving the scope of reason.
Every new discovery started with a question and letting go of some assertions.
https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m...
If going for a pithy answer, I like tha following quote by G. H. Hardy:
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
I think I originally saw this quote in "A Mathematician's Lament" [0] (also known by the author's name as "Lockhart's Lament"), which has a lot to say about modern math education.
[0] (PDF) https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
Imagine that you're creating computer game - a limited simulation. You can change laws of physics, chemistry, life, etc... but math will always hold true. 1 + 1 will always be 2.
Math, as we know it, is invented. It is one set of rules that we like to use to describe things. It is not the only way. I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.
Consider modeling the game of tic tac toe. You can prove that certain strategies always win. If you then proceed to play it with that strategy against the aliens, then you will certainly win the game, regardless of how their logic looks. (this assumes you're starting of course)
That just means proof is no different to an arbitrary set of rules defined acting like a compiler/interpreter for a programming language.
So just Infinite monkey patched classes of definitions... fun.
https://en.m.wikipedia.org/wiki/Chinese_room
Said another way, if you have a dictionary translating, it doesn’t mean the definition is equivalent. In fact, most definitions aren’t, as each one has its nuance. If following a dictionary to arrive at a result after some steps, it doesn’t imply anything other than a set of rules being followed to arrive at an arbitrary value.
Also interesting is the perspective of mind: https://en.wikipedia.org/wiki/China_brain
Universe as one mind depending on zoomed out vs zoomed in.
However this attribution of mind is simply anything that follows rules. So it’s better to call it “logics” and humans have a property of “bio-logic” abilities. But then what exists outside of logic that allows arbitrary definition of things to represent said logic? The programmer, the consciousness. In general I feel mind exists outside of logic but that’s perhaps a matter of perspective and definition. It gets hard to know where one has defined a conceptual space between continuum and borders and assigning it a range of values.
Rules are not what they are because a rule doesn't have any intrinsic property outside of imagination. It's dependent entirely on consensus and memory.
Without communication, consciousness wouldn't be able to distribute rules. It'd be limited to one mind per set of rules. Communication allows imaginary network effects. Distributed protocol message passing, yay!
https://en.m.wikipedia.org/wiki/Chinese_room
Weird. Are we just playing games with ourselves? One programmer existing as the programmer of their reality fooling themselves into perspective?
That's what the aliens could potentially do as well, but that would also be math.
So maybe we just need yet another word? Discovented?
The universe exists as a geometry. That geometry has properties with different interactions. The geometry exists regardless, but we can give that geometry a name arbitrarily.
It’s also the difference between music and math which is equivalent to me as frequency and quantifying frequency.
Vibrations interact with each other. Mathematics is the observation and labeling of interaction(s).
Tricky because vibrations are continuous and now we’ve got a can of worms on continuous and non continuous numbers. Haha.
Tic tac toe doesn't have a forced win though ...
“A 3×3 game is a draw. More generally, the first player can draw or win on any board (of any dimension) whose side length is odd, by playing first in the central cell and then mirroring the opponent's moves.”
In the end does it really matter though? Is it maybe not just that inventions and discoveries have more things in common than some intuitively think? Tons of people are not really found of patents, or at least of some class of patents, because the line is arguably blurry (so yeah there is a practical way in which case it do matter, but only because of a completely arbitrary social organisation)
That isn't an approximation. It is a core principle, not just of math, but of the laws of our universe. It is so core, that it's impossible to imagine a universe without it.
or maybe it is an approximation. Maybe there is some essence of the universe that gives is additiveness - and maybe if you could see closer/bigger/faster, maybe that too would break down.
We should take nothing for granted.
1 river + 1 river = 1 river
1 stick - 1/2 stick = 2 stick
But try the first two with the following substitutions:
"drop of water" => "volume of a drop of water"
"river" => "flow rate of a river"
And you're back to addition and correct answers.
The point mostly being, any symbolic representation is only that. It is never universal. And to think it’s an objective representation of anything is serious error.
2 = 2 is true
2 ≠ 2 is false
1 river = 1 river is true
1 river ≠ 1 river is also true under some circumstancesMath is our reality, and not a universal reality/law/principal (while science conservation of energy/matter is universal). There are no addition/subtraction happening in the universe.
Addition is not conservation. But that's how we see it.
Then as 1 and 1 totaling two rivers they come together then exist as 1.
Representation is relative on an arbitrary border of quantity. The rules leak for any rule that exists.
One father and one mother have a kid. One and one make another one totaling three.
• and • make •.
Usually a person will substitute 1 for the observation of a “thing” so • will be father, and another • mother +1+1=+1
One unit and another one unit makes a third unit. Yet one unit plus another unit is two units.
Not really. If I look at a map and see the Amazon river, and then see the Nile, Amazon + Nile ≠ 1 river. They're still two separate rivers. Adding rivers means merging sets of rivers under consideration, not merging actual rivers.
All you're doing now is playing with different operations that are not what the symbol "+" stands for.
All definitions are nuanced and subjective based on path of least resistance heuristics.
It’s often miscommunication is happening because expected results are misaligned despite using same rules/language.
It’s quite complex. It’s really interesting but also leads to poor communication. Do we try to embed in our cultures to align on expectation and values to improve this or always doomed to potentially fatal inconsistencies? I guess the dynamics of any rules naturally incentivize behavior whether it’s negative or positive.
1 is an abstract concept applied to many things that share this "singleton property".
Nothing is equal in universe. Everything is its own thing. 1 drop of water != another 1 drop of water. Two different entities.
Only in math we have 1=1 or 1+1=2. And math doesn't work for universe, it works for science.
There’s different ways to count based on the dynamics of the system you’re interacting with.
One interaction interacting with another interaction can create a third interaction causing the system to interact as as pieces totaling 3 that came from a 1 and a 1.
1+1=3.
Also, my brain may be split into two spheres however is it better to reason about it as 1+1=2 me’s or 1^2? Funny joke if I use i^2 then I’m just imaginary. Hah.
Ultimately anything we describe as “is” isn’t anything other than agreement of some type of axiom which requires consciousness / imagination.
Lots to ponder. Ha.
If we can't agree on what the words mean, we can't really have a conversation.
More or less consistent enough is certainty.
Nuances exist, as well as unknown side effects.
When has any definition you thought were fixed expanded?
It’s a weird space.
When I figure out I have misunderstood something, it's not the definition that changes. It's my map of it. The map is not the territory.
Setting 1+1=2 creates patterns that often useful in real life (e.g. counting discrete objects). But it's by no means some universal law.
In this case setting 1+1=0 creates other patterns useful for error correcting codes.
Now, of course the symbols of math are invented. But math is the study of mathematical _objects_, and it’s very unclear if those are invented or discovered.
For example, the Curry-Howard isomorphism shows that math proofs and computer programs are the same mathematical object. Does this mean that they are a part of nature, and we discovered them because we invented a sufficiently powerful mathematical system? Or are they simply properties of the invented system?
We do not (and most likely can not) know the answer to that question. If someone is extremely convicted when offering an answer to it, consider me completely suspicious.
We use math based on the status of a sea of transistors to store it in other atoms with forces we're slowly discovering how to communicate about. It's all layers of finding new ways to describe something that was already there, spinning and bumping photons long before any of us had photon detectors in our heads or a chemical machine capable of deciding to call one pattern A and all the ways to translate it into sound waves.
One pattern is called "solid state drive," and we seem to have some say in what that pattern does from our frame of reference. Maybe it was always a solid state drive and we're the language the universe uses to describe its smaller parts.
I promise I'm not high, and probably not a Boltzmann brain. Though I'm not sure about the latter.
It seems fairly clear that the as yet uncalculated digits of pi are there to be discovered rather than waiting for someone to invent them.
We must do better! Good link for anyone really.
But if it is invented then could we not have avoided inventing some of the more gnarly bits? If math is invented then, say, irrational numbers were a terrible mistake.
The building blocks of math are entirely abstract, but once we define a circle in any half-satisfying way then all of the results in algebra, calculus and geometry are largely set in stone. After the invention of the circle in pre-written-history, most of the math that followed was discovery of the implications.
Rational numbers are arguably nice and easy. But if all we have are rational numbers, 2 doesn't have a square root. More in general, our number line has holes, and one of the most fundamental properties of analysis - that every non-empty bounded set has a least upper bound - goes away. We could do away with that (and arguably, irrational numbers don't really exist "in the outside world") but it would make all the calculations so much more complicated.
No model can ever represent nature. A model helps prediction which helps convergence to certainty through consensus.
Although this opens a can of worms of what “accurately represented” means. A threshold comes into account. Then it unpacks into what is existence.
Fun stuff...
I’ve been thinking a lot of “errors” in our models (language, ideas, etc) are emergent from not considering change. Nothing in the universe is still. Everything changes or vibrates. The only things that are still, don’t exist.
It makes me imagine that we need an evolution in our language and core reasoning that intrinsically embeds change and or perspective (relativity?)
I’ve had this thought recently again based on studying what numbers are. The definition of numerals vs numbers vs integers and number theory, etc...
Even the “numbers” that we try to reference seems to exist as non fixed value of infinite change. Also makes me think of what is that core “thing” that allows a number to expose itself within our frame of reality? It seems to consistently show itself through mathematics and information theory and computer science. Perhaps all related to geometry in some way?
but can you imagine such a set of rules?
(Edit: I am getting downvoted, so let me expand my answer a bit.
Nature is explained using Physics and Physics uses Mathematics for its models of Nature.)
This is part of Godel's discovery. Especially, his first Incompleteness Theorem.
https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...
That is to say, even if the incompleteness theorem hadn't been true, math still wouldn't have described nature with perfect accuracy.
Not to mention, the incompleteness theorem hasn't prevented any kind of modelling in physics that I have ever heard of. Indeed, physics is often not entirely constrained by formal methods, with ad-hoc mathematical models that can be shown to work even if they are not fully formalized sometimes being preferred (such as the Dirac delta "function").
Mathematicians use even fewer symbols to say things subject to even stronger internal constraints, so one must pay even closer attention than when reading poetry. (Indeed, almost all people write prose around the symbols as a guide.)
Some poets explore what happens with more, some with fewer, constraints.
Some mathematicians explore what happens with more, some with fewer, constraints.
(Viewed as a MMO game, mathematics introduced the parallel postulate in or before 300 BC and although many people said it ought to be nerfed, the nerfing wouldn't happen until 1830.)
You know this is Category Theory I assume? I guess it does describe math abstractly, but in a way it is also math. Not sure tbh.
If an idea describes reality does that make it a discovery instead of an invention? What if you think the idea describes reality but it really doesn't.
For instance, let's use the sense of inventing as pulling something out of thin air, and discovery as growing an understanding of what is in front of you.
Math certainly involves elaboration according to fixed rules which could be seen as discovery. It also involves what-if anxioms that redefine the basis for the mathematical thought (say 4-dimensional space), that could be thought of as invention.
This article seems to think of invented and discovered in terms of whether it is not based on reality or whether it is based on reality. This is trickier, I think it is fair to say that the needs of reality do drive certain branches of mathematics, but certain branches are proudly not based on reality. The funny thing is that periodically we realize that the math we thought was firmly not based on reality ends up describing reality better than the math we based on reality.
So you have math that was spun out of an idle what-if thought that ends up describing with high accuracy an important physical concept. Is that an invention or a discovery?
That statement surprised me!
Google told me that according to this biography by J J O'Connor and E F Robertson [0], Roger Penrose in a debate with Stephen Hawking in 1994 self-described as a realist rather than a Platonist. They quote Penrose as follows:
>At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions.
[0] https://mathshistory.st-andrews.ac.uk/Biographies/Penrose/
Can you touch concepts? Can you make an object fall upwards against the laws of gravity in your imagination?
The best definition I've heard calls math "the science of pattern". However, proving itself undefinable seems fundamentally better.
The other one was “why were the Greeks (and contemporaneous Indians) wondering about geometry of a plane and why were the Arabs thinking of algebra” when they had more limited technology than we do?
there's just too much busy-work in modernity
TL;DR: they were curious, they saw some patterns in their daily experiences, and tried to generalize them.
So the first thought I formulate is an empirical proof that reality has structure. Mathematics is nothing but a description of that structure of reality. We may describe it in human-centric terms and models, but the structure must by necessity pre-exist the mind and is empirically discovered by the action rational tought, not created by the mind.
No similar conclusion can be drawn for other non-empirical hypothesis, such as the existence of God. My ability to think does not imply the existence of anything else beside myself, so other empirical proofs must be considered.
To me, a proof is a deductive mathematical argument that takes a claim of the form “if P then Q” and shows how Q is necessarily true when we assume P. I don’t know how to translate from reality into that formalism. All of modern mathematics does not attempt to do so. Instead, it is built up from the axioms of set theory, which take the form of rules for manipulating symbols on paper (or on a computer or even in our minds) without saying anything specific about what those symbols really mean.
That still leaves the definition of "real" and "exists". And even if a precise definition is given, that leaves the question of whether you are real or not. The answers to these questions are not incidental and are indeed relevant to your subsequent assertions.
Why must the structure by necessity pre-exist the mind?
In other words, you are asserting that idealism (all reality is mind-dependent) is necessarily false, but what is your argument to justify that assertion?
(Even someone who believes that idealism is false might not agree that it is necessarily false, although the term necessity is ambiguous and can have different meanings – broad vs narrow logical necessity, metaphysical necessity, epistemic necessity, nomic necessity, real necessity, etc.)
While it can be used to describe the reality we experience about our universe, it can also create imagined universes that have only the most tenuous grasp (if any). Like any language it supports fiction and non-fiction applications.
There are also rules about how the language evolves that are more strict and defined than human language and the evolution often is more akin to discovery. It’s like the exercise to try to write every single possible English language novel to find Shakespeare. Except instead of stories about human experience we’re trying to identify which random storyline we’re following maps to some fundamental truth about how the universe works.
You can sometimes also map fiction to our experience of the world so that it becomes more tangible (eg playing with 4d objects in a 3d video game viewed through a 2d plane).
This is no different from reading a fiction book to learn about concepts like compassion even though the story itself is made up.
So it’s not either or to me. Both empiricism and Platonism interpretations coexist. Empiricism is what we use to understand what story reality is telling us and how to confirm our comprehension of that story is correct. Platonism is that mathematics itself as a concept is something humans can explore and try to master but exists outside of us and the rules of the language are predefined outside of humanity. Note that just like any language it even has really significant gaps you can’t do anything about (which ironically the language gave us the ability to prove that about itself).
That being said the original question this article references indicates a lack of curiosity, which is where my problem with it stems from. And it’s not even like it’s hard to find (although it may require more than a superficial query). “What are practical applications of algebra” is answered in the first few hits. “Who discovered/invented algebra” can be similarly answered. These questions only sound deep if you are actively avoid actually trying to find answers to these questions. There are legitimate philosophical discussions to be had but the original question was not an attempt to have that discussion.
Maybe I am wading into well-charted waters and thinking of something others have already hashed out, but if our universe only applies a subset of math, could that also mean there is another universe where a different subset of math is applicable? And the multiverse in sum encompasses all math?
Platonism would then be true at the most global scope, but empiricism is necessary to discover what part of math our particular universe encompasses.
Math can't tell you that there are three of [anything that you encounter with your sensory organs], but it can tell you that if there are three of those, you can also say...
In that way, it tells you about the real world while having no connection to any real world. The problem is that as soon as a math-predicted observation fails, we can blame the mistake on our sensory organs - either formatively (when we assigned a particular observation to a particular mathematical entity (like "three"), or conclusively (when we assigned a mathematical result to a sensation.) If a mathematical result consistently fails to apply to any sensation, it ceases to be math - not by disappearing, but by failing to be consistent; if you can't apply a mathematical process to a sensation in a way that enables you to predict a subsequent sensation, there's no way to determine whether the math makes any consistent statement.
Math has to be empirical because it's completely vetted by sensational observations. If a hypothetical god changed some aspect of the underlying physical reality (inclusive of changing some aspect of human sensory organs), the math would change along with it, and things that were math would cease to be math. If you included that god within your mathematics, math becomes the set of all possible maths, which is just the universal set. There's no meaning to a math that says everything about everything.
I guess you can say Platonism is true too, since math is a statement about an underlying reality (because it is discarded when it is not), but it's not a meaningful thing to say. Math that turned out to be correct (and continues to turn out to be correct) is Platonic. Math that was incorrect (or in the future becomes incorrect) is not Platonic. The only way you can determine which math is Platonic and which math is not is through empirical observation, and the degree to which you can be sure that particular math continues to be Platonic is through how well your past observations predict your future observations.
To answer the tweet, people study math because because being able to predict things gives you an advantage in your work.
I also think that the fascinating part of programming is that it's sitting in between math and fiction, depending on an application. Sometimes you have to obey the rules very strictly (e.g. writing a parser), but in other times you can be wildly imaginative (e.g. designing UI).
This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist?
I'm willing to admit that concrete instances of both exist in space and time (and I might even admit maps forming a galois connection between those and the abstract ideal 'A') but there's no "reference" canonical A, located in time, located in space, giving them meaning.
Only in our minds ? A doesn't really exist IMO.
Logic is programming. Human’s are programmers and hardware/OS who largely don’t realize they’re running arbitrary truth values based on popular imaginary rules.
To exist, is to LARP? Lol.
I like to think of mathematics as the formalization of human intuition. Insofar as that intuition has been shaped by our observations of the physical world, we should not be surprised that many of our mathematical models seem to describe aspects of natural phenomena.
[1] https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/...
He also tries to politicise the debate which I feel is kind of silly.
Math is the language of the simplest definitions.
Math is the language of the most useful definitions.We don't know much about Pythagoras and why he did mathematics.
But we do know a bit more about Archimedes and the reasons why he did mathematics: to kill Romans and improve plumbing.
Two things that mathematics was found to be extremely good at.
To start off, math is simply the art of understanding the consequences and theorems from a set of axioms. Since there are an infinite amount of sets of axioms, there's an infinite amount of math.
Whether a set of axioms or the theorems derived from a set of axioms is discovered or created is again a pointless philosophical debate that is not exclusive to mathematics. You can apply this question to any topic. Case in point:
There is a finite set of paintings that can be painted on a fixed sized canvas using a fixed set of colored paint. When you create a painting... are you creating a painting or are you "discovering it" from the set of all paintings?
Since the question does not pertain exclusively to math; if you think hard you can sort of realize that it's just philosophical speculation derived from intricacies in our language. Is anything truly created? Or is it just "discovered" from the set of all possible things?
When we think about this stuff we're not actually thinking about anything profound, rather we are just examining intricacies of language and the definition of the word "discover" and "create." I mean if you think really hard and try to formalize the definition of "create" it can be said that to "create" something is to "discover" a single member from the set of all things. It can also be said to "discover" something is to "create" a new member from the set of all things. People are just getting excited over a language issue. The word "create" was never formalized and forms a sort of fuzzy overlap with "discover."
Basically it's one and the same thing sort of like is the glass half full or half empty? The actual reality of this problem is uninteresting and pointless. This paradoxical-like phenomenon doesn't actually exist and arises exclusively because in our language, we have a word called "empty" and we have a word called "full."
Consider the black wall. Is the wall "black" or does it lack color? Consider the flying bird. Is the bird flying or is it falling and missing the ground? Is the dark room full of darkness or is light just absent? It's all just language tricks I can "create" or "discover" an infinite amount of garbage here for you to think "profoundly" about without actually realizing it's just a language issue.
We like this particular set of axioms and rules because they can answer questions about physics, but they were stumbled upon almost by accident when we started using the concept of numbers and asking "What does it mean to prove something?".
One could imagine other points in this space though, which define, say, the game of chess, or perhaps something as complex as English grammar. It just so happened that the point we naturally settled on for mathematics is one that was powerful enough to lead to the rules of algebra and geometry etc., and also consistent/deterministic in the same way that most physics seems to be (which is convenient for humans, whose brains are limited by physics).
So, mathematics doesn't "underlie" physics in the sense of being more real than it, but merely is a point we have chosen because of its similarity to physics. I suppose the real question, then, is why should such a useful point exist in this infinite space at all.
No, the word mathematics actually describes this entire space. You can go a level below math down to logic. Logic are the rules that bind all these different mathematical fields together.
In addition to axioms that are assumed to be true, Logic in itself is also assumed to be true and we use a singular set of logical rules for all of mathematics.
>We like this particular set of axioms and rules because they can answer questions about physics, but they were stumbled upon almost by accident when we started using the concept of numbers and asking "What does it mean to prove something?".
Much of abstract mathematics has no real world counterpart.
Put this way. Whether something was "created" of "discovered" are human concepts materialized through language. To apply it to math is pointless.
Did we discover the decimal form of representation out of all possible forms of representing fractional quantities or did we create that form?
Was the english language discovered out of the set of all possible languages or was it created?
It's good to examine the intuitive differences though. Why does the choice of representation feel more "creative" than say the "choice" of what axioms to use in a certain field in mathematics?
Many times you will discover that the intuition is just a biased flaw in the way we think, but many times you will discover that your intuition is demarcating an actual difference.
I think our intuition is marking an actual difference here. The set of all symbols representing the number 3 is infinite not bounded by any rules. The set of all sets of axioms we can use to formulate a mathematical theory is infinite but it is actually a subset. We only choose axioms in math that are internally consistent. The number Inconsistent sets of axioms are a huge and many are discarded to find a consistent set.
It is possibly this blurry demarcation of restriction and size of the domain that marks our intuitive feelings of what we label as "created" or "discovered." Who knows, I suspect the rules are much more complicated and if we dive in deeper you will find that our intuition is flawed.
For example why does it feel extremely wrong to say that the Americas were created rather than discovered?
But again, realize that we are discussing the intricacies of arbitrary words defined in language. Outside of the domain of language the concept is uninteresting.
The case of mathematics seems more subtle though, beyond language issues. I find it quite interesting myself.