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.
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.
As the other commentator points out, "Continuity and Limits" are fundamental to explaining physical phenomena (via differential equations) and are not "purely mathematical constructs".
You point to the argument that "we have to calculate with them, therefore they are real", but another aspect of the debate is about which mathematical entities are real in the sense of being physically realizable. From this perspective, it's not even clear that real numbers are real, e.g. whether and how it would physically possible for a real-number based quantity to be present in a finite space.
If Mathematics helps in explaining Nature/Universe then it is "Real" no matter if the explanation goes through a whole stack of abstractions or not. There are two main parts;
a) A Formal System consisting of Set of Objects, Operations, Mappings, Axioms and Logic Rules. We invent the symbols and notations to express these.
b) A Domain of Discourse/Interpretation in which the above is applied to map to "Reality".
We Humans have an innate sense of Quantities, Proportion, Objects and Relationships which is what can be called the "Mathematical sense". Even the most uneducated goatherd can count his goats (eg. using pattern-matching with one stone per goat) without knowing anything about the number system. He can also compare his bunch with his neighbour's and tell you which is larger. If you throw a ball at him he can estimate its trajectory and move accordingly to catch it. We have merely abstracted out the essentials from the above and modeled them as Set Theory, Integer/Real Number lines, Rate of change of one quantity w.r.t. another etc. and labeled these as "Mathematics". The models are by design "abstract" but once applied to a "domain" become concrete.
I agree with everything you say but that's not really what I was talking about.
Physical realizability concerns the question whether some type of entity can in principle exist in the known physical universe, whether it's physical existence would violate existing laws of nature. The question is independent of the question whether there is (also) a Platonic realm of mathematical objects (although there is a connection if you are neither a constructivist nor a Platonist). As far as I know, nobody doubts that integer quantities can be physically realized without violating existing laws. Likewise, you can say that a square is an abstraction from a square macroscopic object, even though no side of that object can be perfectly square in nature.
However, the case with real numbers is a bit different from the square. It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if it's not an abstraction from something that clearly can be physically realized, then it is meaningful to ask whether a real-number quantity can exist in the physical universe. From what I remember, some philosophers and physicists think the answer is No.
>It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if it's not an abstraction from something that clearly can be physically realized, then it is meaningful to ask whether a real-number quantity can exist in the physical universe.
Real numbers are absolutely "physically realizable" (in the sense that you are defining it) in the Physical World. If you have 3 litres of water and you give me half, you have just "realized" the Real Number 1.5 from a "quantized integer" 3. Incidentally even integers are just an abstraction of attributes of collections of things i.e. cardinality of a set of things. This is why i tell people to look at Mathematics as a Formal System+Domain of Discourse in the Real World. You do all your symbol manipulations in the former and at the end map it to the real world to see whether it is valid.
You're right that some real numbers are physically realizable, but not all of them. The debate usually focuses on irrational numbers. Please bear in mind this is an existing debate in the philosophy of mathematics, not my personal invention, and I have to apologize for being somewhat vague about. I tried to find a paper I've stumbled across years ago but couldn't find it, so I'm writing from distant memory.
Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws.
Now don't get me wrong, this is all controversial. The idea is, for example, that π cannot be physically realized because it has an infinite decimal expansion. Some people would agree, other would disagree.
I understand why you disagree, but bear in mind my original point was not to argue that real numbers aren't physically real, but rather that there is no general agreement about this issue among people who muse about these kinds of philosophical questions. The question is relevant for foundational views about mathematics. If certain real numbers like 1/3 and π cannot be physically realized, they cannot be abstractions from something encountered in nature (at least not in the sense of "abstraction" according to which some properties are ignored). The view remains compatible with regarding them as mental constructions and compatible with mathematical Platonism, though.
The difficulties with Irrational Numbers were obvious from the beginning (https://en.wikipedia.org/wiki/Irrational_number#History). Many of them are very "Real" and occur naturally in the Universe (eg. Pi = Circumference of a circle / Diameter of the circle) but our modeling of them doesn't feel "natural" and hence the confusion. You might find the following interesting;
> Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws.
Well, Yes, that's what the debate is about. Just to make this clear (in case someone else reads this thread), this is an ongoing debate in the philosophy of mathematics and not just something people muse about on Quora and Stackexchange. What's important for me is that the notion of physical realizability is understood correctly. It does not pertain to philosophical arguments or common sense, it really means that something can be present as a quantity in nature (actuality) without violating existing physical laws. When someone argues that a mathematical structure or entity cannot be physically realized, the argument must concretely show that the realization would violate some currently well-confirmed laws of nature or fundamental physical principles.
There is no disagreement with you. I just wanted to clarify that (hope you don't mind). I didn't have just any arguments against irrational numbers in mind but a specific type of arguments.
Your idea of physical realizability seems to me to be questionable. Because there are many abstractions in a chain starting from mental concept to a final physical object where each link is necessary to get to the final result (eg. a modern computer). Do not get caught up in metaphysical arguments of philosophers on mathematics which are often just playing with words rather than substance (eg. my link above to God created the irrational numbers by a PhD in Philosophy).
Finally to conclude this thread; i highly recommend reading The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner if you haven't already done so.
> your idea of physical realizability seems to me to be questionable.
I made it abundantly clear that is not my idea but an ongoing discussion in the philosophy of mathematics. You're telling me to not get caught in philosophical arguments and in the very sentence before that presuppose the idea that mathematics is a mental construction, which is just one out of many philosophical views in that area. By the way, I'm interested in philosophical issues because I am a philosopher. Just because you don't like these issues or find them "questionable" doesn't mean anything. Some of my colleagues defend an Anti-Fregean formal foundation of mathematics to which physical realizability seems to pose a huge problem. If they want to get their papers published, they'll have to address the issue.
I'm aware of Wigner's paper, it's a well-known classic. Finally, to conclude this thread from my perspective, while I'm personally not interested in the (broadly conceived) metaphysics of mathematics and am happy to leave these issues to mathematicians interested in them, the way you're just presupposing that mathematical objects are mere mental constructions cannot really count as engaging with the problems yet. If you read what I wrote above again, you'll realize that I presented an argument why irrational numbers cannot be abstractions, yet you keep talking about abstractions. In a nutshell, it's not that simple.
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.
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).
I’m not disagreeing just thinking your line of argument must say more about where all this perpetually undiscovered math is confined in the universe. It sounds like you’re forced to believe math is abstract in nature then, but doesn’t that conflict with your earlier points.
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.
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.
"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
