I understand you point of view: I think this was the common point of view of mathematicians from ancient Greeks until the beginning of XVII century: you cannot allow the concept of actual infinite (and infinitesimal) in mathematics because from it you can derive paradoxical conclusions (Zeno's paradox, for example), and if paradoxes are allowed you cannot trust mathematical proofs any more.

But then people started to discover that you can use infinitesimals to derive lots of correct results about the laws of nature (law of gravitation, for example). So, there had to be some way to allow the calculations that gave correct results using infinitesimals!
The solution to the problem was to define real numbers as "limits" (Cauchy) or "cuts" (Dedekind). But, in both cases, they are made of actually infinite objects (infinite sequences or infinite sets). So, at least for what I know, if you want calculus to be part of mathematics you have to allow the existence of actually infinite objects!

Formal logic and set theory were mainly created as a way to reason about the counter-intuitive concepts of "infinite set of things". If you do not allow infinite sets, set theory becomes trivial, and fundamentally useless: you don't need an axiomatic set theory to reason about finite sets.

With formal logic, you don't need a "concrete model" of mathematical things: a "set" is whatever "thing" on which you can operate following a (carefully chosen) given set of rules. So you can reason about infinite things avoiding contradictions.

But in this way, you cannot say that "this thing should not be allowed because it does not make physical sense". Everything that is not contradictory should be allowed, since we are not speaking about physical objects any more.

For what I know, there is no other way to make sense of calculus or differential equations (and other useful concepts in mathematics) without using formal logic or without "concrete" infinitesimal objects.

Well, not exactly...

∞ - ∞ = 0 is not true, because ∞ - ∞ is not a term of the language.

Let me explain better: let's take the function "int", defined as "the integer part of a decimal number":
- for example, int( 5.65 ) = 5
I want to invert this function, and call it "unint", such that unint( 5 ) = 5.65
I cannot do it, because int( 5.7 ) = 5, and int( 5,8 ) = 5, etc..

This is the same situation:
I define ∞ + 0 = ∞, ∞ + 1 = ∞, ∞ + 2 = ∞, ecc..
So, every finite number is a neutral element for ∞, such as 0 is THE neutral element for natural numbers.

Now, if for transfinite numbers the neutral element for addition is nor unique, you can't define subtraction as the inverse of addition, simply because addition is not invertible!

If you believe there is only one infinity (like I do) — Devans99

Sorry to disappoint you, but (as I just wrote in the previous post), I believe that at least infinitesimal objects do exist in nature in some sense, but you cannot decide if they exist or not using mathematics.

From the point of view of mathematics, I don't think there is a reason why transfinite numbers should be not allowed. Yes, you can say "I don't like set theory because it's a complicated way to define something that seems intuitively simple as sets", but a hundred years ago this was the only theory that allowed to reason in an unambiguous way on real numbers (that, as the name suggests, appear to be "real") and to say such an obvious thing as the fact that the square root of two (i.e. the measure of the diagonal of a square whose side has measure one) is a number.

And without real numbers how do you build limits and calculus? And calculus has to be "real", since it works so well with the laws of nature.

So, my point is that you can use logic to speak about infinity even if you cannot decide from logic itself if infinity really exists in nature.

If he said there are finite models I'm sure there are! — fishfry

First of all, I used the term "finitary model" meaning "a model that is built from a finitary theory", but probably it's a misused term. So, let's speak only about "finitary first order theories" with this definition: "A first-order theory is called finitary when it's expressions contain only finite disjunctions or conjunctions".

Of course I could be wrong, and I could have misinterpreted what Voevodsky says in the video, but the fact that first order set theories have a model whose elements can be put in a one-to-one relation with natural numbers is a theorem of model theory, basically due to the fact that you can always take as a model the strings that correspond to well-formed terms of the language: in other words, the theory can be interpreted as speaking about "strings of characters".

As usual, a citation from the old good wikipedia can come to the rescue :-)
"Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model."
(https://en.wikipedia.org/wiki/Model_theory)

So, you could even say that "every mathematical object is a string of characters", or (as Godel did in his famous incompleteness theorem) that "every mathematical object is a natural number" (because he proved that every theorem can be translated in another theorem speaking about natural numbers).

This of course doesn't mean that uncountable sets "do not exist", but only that you cannot use a finitary
first order logic theory to prove that they exist.

But, at the end, you find that every formal logic based on rules and axioms has the same problem (I know, I should explain why, but it would be too long to start discussing this here). For this reason, I believe that the existence of non numerable infinite sets should be treated as a problem to be decided by physics, and not by mathematics.
I know the obvious objection: how do you make a physical experiment to check if something (like for example space-time) is made of a countable or uncountable set of points? Well, I don't know :-)
But the fact that calculus becomes much simpler and "beautiful" if it is interpreted using infinitesimals instead of limits (look for example at smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis) for me is a good indication that, in some sense, infinitesimal "objects" exist in nature, because laws of physics are written with differential equations.

If you want to use formal logic (that, for what I know, is the only form of logic that we can trust for sure), you have to define the term "objective reality".

If "objective reality" means mathematical model, as you said, this sentence is trivially false: it is false that there is no way to show that a proposition is true in all models. In fact, all mathematical theorems that have a demonstration are true in all models.

But probably by "objective reality" here you mean "physical reality" or "physical universe". So, the sentence becomes "there is no way to show that a proposition is true in the physical universe". In this case it depends of what you mean by "proposition" and "to show".

If you interpret "to show" as to perform an experiment and "proposition" as a description of a physical experiment, than the sentence is again false, because it is possible to perform experiments in the physical world with a result of "true" or "false" (only that it is not guaranteed a priori that the same experiment gives always the same result).

But if you want to interpret "to show" and "proposition" in the mathematical meaning (as a purely formal system of rules based on axioms) and you want to take as model for the variables that appear in the proposition objects of the physical world, then in my opinion the proposition is ture: you cannot make demonstrations using formal logic referring the variables to objects of the physical world: the porposition can be true or false depending on which particular object of the physical world you are referring to, but there is no way to specify to which physical object you refer using a formal language. So, every proposition interpreted in this sense is not demonstrable, meaning "not true in all models".

This is my opinion, but probably I missed something important because i didn't read all posts from the beginning.. :-) Please say me if I missed something important.

Yes, exactly.

The world and our mind have forms that are similar to interesting mathematical objects because these forms are in some way special, and the laws of physics favorite the development these forms (i.e. forms with an high degree of symmetry) respect to the others.

But why did we choose numbers and geometrical objects for our mathematical exploration? Stepping back to an abstract remote once again, they are nothing but mathematical constructs - a drop in an infinite sea of such constructs. There is no a priori reason to favor those concepts over any other. So the reason will not be found in the abstract enterprise of mathematics, but in the world that we inhabit, and perhaps in the contingencies of our cognitive and cultural evolution. — SophistiCat

This is a post from 5 days ago, but it's an interesting subject so I'll reply to it now.

I think that we could find the reason to favor numbers and geometrical objects over the infinite sea of other possible mathematical constructs "in the abstract enterprise of mathematics", without looking at the world that we inhabit. For example, we could find a mathematical function that takes as an input the formulation of a mathematical theory in some formal language and returns a positive number that is a measure of how "interesting" is that theory. The function should be made in such a way that when it's given as input mathematical theories generated by putting together axioms and rules at random, the return value is very low. Instead, if t's given as input mathematical theories that we judge as interesting, the return value is much bigger. This would be a judgment independent of the physical world, "internal" to mathematics itself. I think that it wouldn't be difficult to build such a function with one of the technologies used today for pattern recognition algorithms, such as neural networks. The problem is that to teach the neural network which theories to recognize, you have put them inside yourself, so in reality it would be only a memory of the theories that we judged as interesting. But there is an objective way to decide if the function is only a memory of the things that we just know, or something more: the quantity of information necessary to describe the function should be much smaller than the quantity of information required to write the theories that it recognizes.
Well, in my opinion the theories that we find interesting differ from the others for the fact that we can "encode" a great quantity of apparently unrelated facts from a very small quantity of rules and axioms. In other worlds, they are full of symmetries, and that's basically the reason why we find them "interesting".
Well, in reality I am not convinced that the thing is so simple as I described it. Probably that function would be impossible to calculate even if possible to define, or would not agree completely with what we judge "interesting", but I think that something like this will be discovered one day in mathematics.

↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:

∞ + 1 = ∞

This assertion says in english:

’There exists something that when changed, does not change’

Thats a straight contradiction. — Devans99

I read the book that you suggested me (thanks for the link!)
I think that the "trick" that avoids inconsistency is that the inverse of infinite is not unique.

From wikipedia (https://en.wikipedia.org/wiki/Cardinal_number):
Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

So, choosing μ = σ = ∞, there exists a cardinal k such that ∞ + k = ∞
k will be unique if and only if ∞ < ∞. But ∞ < ∞ is false, then k is not unique.

So, it's impossible to deduce that ∞ - ∞ = 0, and it's not allowed to use the expression ∞ - ∞ as if it was a definite cardinal number.

To derive 1 = 0 from ∞ + 1 = ∞ you should subtract ∞ from both sides of the equation, but
"1 + ∞ = ∞" does not imply "(1 + ∞) - ∞ = ∞ - ∞" because ∞ has not an unique inverse.
So, no contradiction! :-)

By the way, "∞ + 1 = ∞" does not mean ’There exists something that when changed, does not change’, but 'there exists something that doesn't change when you add 1 to it'

For me, if math is a discovery then it must exist in the world ouside of our minds and that, to me, points to the mathematical laws of nature (science). — TheMadFool

I'll give you an example: the game of chess exists "ouside of our minds" as list of possible "positions" and a list of allowed "moves" that the players can perform to pass from a given "position" to the next one. It doesn't matter what physical model you use to represent the positions (usually the chessboard and the pieces) and in what form you write down the set of rules (as soon as you are able use them to decide in an deterministic way if a given move is allowed or not).

So, in whatever "thing" that allows to distinguish objects from one another and to build rules that can be followed in a deterministic way, the game of chess exists.
Our mind is one of those "things" that allows to represent the game, so game of chess can exist in our mind. Our physical world is another "thing" that allows to represent the game. And there are plenty of "things" in which the game can be represented.

Now, the point is that in all of these "things" the game of chess works exactly at the same way. So, if somebody proves one day that there exists a winning strategy for the white player, this theorem is not about our mind or our world, but is rather about all "things" that have the ability do distinguish objects from one another and to follow deterministic rules.

If there were nothing in the universe with these characteristics, then both the game of chess and even mathematics did not exist. But since there are other "things", except our brain, that have these characteristics, both the game of chess and mathematics exist independently of our minds.

Without a connection to the real world math would only exist in our minds; making it an invention. — TheMadFool

This is the point: it's not necessary a "real" world to represent math, but anything capable of following a set of rules (including a "virtual" world)

Perhaps it should be noted that any computer follows algorithms in a specific way (referred typically as the program it runs), which makes the whole thing quite mathematical. — ssu

Yes, but even if the algorithms were subject to any arbitrary changes by the author of the game, he would still be able to prove the same theorems of our mathematics, if he is able to use the objects of his world to build a model of the theory (probably made of symbols), and then verify that these objects have a given set of properties following the rules of logic. At the end, it would be enough to be able to define an algorithm (made of a given set of rules), that behaves always at the same way when you run it with the same input. If in your world there is no way to define how to perform addition such that the sum of the same two numbers gives always the same result, than mathematics for you makes no sense.

Let's try to make more precise the distinction between "mathematical" and "physical" law:
suppose that our alien was not a real alien, but a character of some computer game, and he discovers that fact (like in film "Nirvana", one of my favorite ones: https://en.wikipedia.org/wiki/Nirvana_(film)). Obviously, in his world there are no real "laws of physics", because he knows that the author of the game, that has full control of the world where he lives, could make happen whatever he wants: objects or persons can disappear, or move at instant speed, and the whole universe that he sees is only a simulation. So he knows that in his world there are no "laws of physics". But could there in his world still be "laws of mathematics" and mathematical theorems? I think the answer is yes! For example number theory is based only on the fact that natural numbers and logical rules are "constructible", and I that is based on a very minimal set of requirements that the "physical universe" must have.
So, with this definition of "laws of mathematics", I believe that it would still be possible to reconstruct most part of what we call mathematics today. So, I think that there is some set of "interesting" mathematical constructions that are different from pure logical combinatorial games in some concrete sense, and are not really related to our particular laws of physics.

↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:

∞ + 1 = ∞

This assertion says in english:

’There exists something that when changed, does not change’

Thats a straight contradiction. — Devans99

Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested!

It is the assumption that infinite sets are measurable that invalidates naive set theory. ZF set theory is patchwork of hacks that tries to cover all the the holes and fails - the solution is to acknowledge infinite sets do not have a cardinality / size. — Devans99

Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not inconsistent!

↪Mephist Sorry, using 2^S to denote the power set of S. The proof I gave is meant to show that the set of all sets does not exist. I maintain that it is the cardinality of the set of all sets that does not exist. — Devans99

OK, I didn't understand at first that you wanted to use it as a "fasle proof". I agree, this proof is surely not acceptable for a number or reasons. First of all, we know from Russell that we should be very careful when proving something about the set of all sets, because the "naive set theory" is contradictory. So we should use a formalization of set theory as Zermelo Fraenkel Set theory, and complete formal proofs in ZF Set theory are not so simple..

1.Let S be the set of all sets, then |S| < |2^S|
2. But 2^S is a subset of S, because every set in 2^S is in S.
3. Therefore |S|>=|2^S|
4. A contradiction, therefore the set of all sets does not exist.

What is wrong with this ‘proof’? — Devans99

The conclusion could even be that "the measure of the set of all sets does not exist" this is an assumption too.

If you use a reasonable definition of infinity: ‘A number bigger than any other number’ then it is clear that there could only be one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity — Devans99

True. But you can't use such number system (integers plus "infinite") to define a "size" of sets that respects the properties 1 to 4.

This is a nonsensical definition: for instance, it claims the even numbers are the same size as the natural numbers (as there is a one-to-one correspondence between the two). But the even numbers are a proper subset of the natural numbers. If either had a size, the size of the natural numbers must be greater than the size of the even numbers. — Devans99

Well, the problem is to give a definition of "size" for sets that you cannot count. And we want this definition to have some "reasonable" properties:
1. it should be the number of elements if the set is finite
2. it should be applicable to both countable and uncountable sets
3. any uncountable set should be bigger than any countable one
4. (as you required) if A is a proper subset of B, than A should be smaller than B

- To satisfy 3 you have to add an extra element (that we call "infinite") to the integers as "size" of uncountable sets, because all integers are just "taken", and by definition we require this element to be greater than every natural number.

- To satisfy 4 you have to add a lot more elements of the "infinite" kind, that have to be all different between each-other.

I don't know if it's possible to define a "size" that respects these four properties, but even if it is, you cannot do it with only one "infinite" size.

If instead you drop the property 4, you can require (by definition) that two sets have the same size if there exists a bijection between their elements. This is uglier, but not contradictory: it's only an equivalence relation between the sizes of the sets. You could even require that all the sizes between 10 and 20 are the same, and all other sizes behave normally, and that wouldn't be contradictory either.
The point is that this is only a definition: the point is not if it is "true" or not, but if it is not contradictory and if it can be used to build "interesting" theorems.

Oh my, no. Not at all. You should read the link you posted. There's no nonstandard finite model of ZF — fishfry

Sorry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets)

However, this idea is not mine: (https://www.youtube.com/watch?v=UvDeVqzcw4k) see at about min. 8:23

Perhaps you're thinking of the kind of set theory used by Frege — fishfry

Yes, exactly. not in "Principia Mathematica" ( my mistake ).

OK, I am glad to hear that you agree :smile:

I'll add even something else, that probably everybody will cause a lot of dissent.. :smile:
In my opinion the existence of infinite sets (such as the uncountably infinite number of points in a line) is not "demonstrable" by logic or mathematics, but is more related to physics. Then, I think that axioms like the axiom of choice or the continuum hypothesis, that are logically independent from the rest of "purely combinatorial" axioms should be treated in a similar way as the Euclid's parallel postulate in euclidean geometry: they are not decidable on the base of the sole logic.

You have claimed that Russell's paradox invalidates the powerset axiom but I still don't follow your logic. In fact if the powerset axiom were false, I would have heard about it. — fishfry

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica

OK, maybe I wanted to make it too simple :smile:
ZF Set theory is another way to limit the use of "naive set theory", in my opinion much more complicated than Type theory.
The point is that both of them are "axiomatic" theories: ZF set theory speaks about something called "set" giving a complicated set of rules on how to operate with them. But with the same set of rules the world "set" can be interpreted as a different finitely defined structure (for example as as recursively branching trees). Every non contradictory axiomatic theory based on first order logic has a finite (non-standard) model (https://en.wikipedia.org/wiki/Compactness_theorem)

From (https://en.wikipedia.org/wiki/Russell%27s_paradox):
"In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory."
Russel's paradox is about "the set of all sets that don't contain theirself", but the problem was how to limit the language (axioms and rules of logic) to ensure that no sentences of the type "the set of all sets that don't contain theirselves" are allowed. One of the solutions is type theory.
In the standard contemporary mathematics based on set theory you can't speak of the "set of all sets" because it's not a set itself, but rather you have to speak about the "class" of all sets.
If you use first order logics on the domain of real numbers, the set of all subsets of real numbers is the same thing as "the set of all sets"

First, we realise if two different types of infinity existed, they would have to be larger than each other. Thats a logical contradiction, so we must of made a wrong assumption; only one kind of infinity can exist. — Devans99

Cantor's "diagonal" theorem on the existence of an infinite hierarchy of infinities can be expressed in a quite convincing way: "for every set A, the set of all subsets of A is bigger than A".
Of course, we have to give a concrete definition of what "bigger" means:
"the set B is bigger than the set A if there isn't any function that for every element of A gives an element of B and covers all B" ( i.e. each element of B corresponds to some element of A )

The demonstration is quite easy (https://en.wikipedia.org/wiki/Cantor%27s_theorem). But there is a problem with the statement of the theorem: Russel's paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox). The concept of "set of all subsets" is contradictory.

So, Russel's idea is, basically, to avoid talking about sets, and talk only about intuitively well-defined things: functions defined on "recursive types". In this way, Cantor's theorem becomes: "for every recursive type A, the type A->Bool is bigger than A".
Put in this way, Cantor's theorem only means that it's impossible to build a "surjective" function that associates to every natural number ( "Nat" ) a second function from natural numbers to booleans ( "Nat -> Bool" ). So ( "Nat -> Bool" is bigger than "Nat" ). At the same way, "Nat -> (Nat -> Bool)" is bigger than "Nat -> Bool", "Nat -> (Nat -> (Nat -> Bool))" is bigger than "Nat -> (Nat -> Bool)", etc..

This is what it means "an infinite hierarchy of of infinite sets each one bigger than the other".
Everything is built using functions on natural numbers and very basic logic deductions, and that's the whole point: it makes sense to speak about infinite objects because they are only built by finitely defined functions. So, in a sense, from the point of view of logic, all infinites are only "potential"

Another way of saying that would be that it's possible for an invention to have a twin in the world. Before we find the twin it's an invention but after we find it it's a discovery. A better example would be, history seems replete with examples, I thinking of a particular theory for the first time (invention) only to find out later that it's an older theory (discovery) — TheMadFool

Well, that's not exactly what I had in mind saying that "the fact that two independent civilizations "invent" the same mathematical theorem is a proof that the theorem has an underlying objective reality". I think that the "underlying reality" of the theorem is there even if it were not "invented" by anybody. But the fact that is "invented" at the same way by many independent civilizations would be a proof of the fact that it is not merely one of the infinite combinations of logical symbols that can be built with the formal "logic game". Of course, we don't have an independent mathematics built by aliens to compare with, so we cannot be sure which parts of our mathematics (if there are) are merely logical games with no other underlying objective meaning. Or maybe there exist some way to "measure" the importance of theorems, but we didn't discover it yet.. ( see https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem ).

Math is like that. It's not entirely a discovery because some math have no application (as of yet) and it's not all invention because some math have real-world applications. — TheMadFool

I think that the fact that there is no application it's not a good clue of the fact that there is no objective meaning

Well, what I meant to ask was is if could exist (or if somebody invented) any "automatic" and "objective" way to recognize meaningful mathematical theorems (or theories) when they are expressed in a formal language.
Probably the answer is NO. But if there is no such thing as an objective "value" of a mathematical sentence, how can mathematicians be able to recognize an "interesting" new theory when they see one?

The wheel example is perfect for the occasion. It informs us that the invention-discovery distinction is not as yet clear to us. I mean we're misapplying the words ''invention'' to wheels rather than that the aliens ''discovering'' math is wrong.

I think math is both an invention and a discovery. As Sophisticat pointed out we create mathematical worlds using axioms of our choice. These remain in the realm of invention until it finds application in the world after which we see it as a discovery. — TheMadFool

I think that we could use quite simple unambiguous definitions of invention and discovery:
-- "invention" is the creation of something that didn't exist before: by "exist" I mean of course that "had not been built", not that did not exist as possibility. For example, a new novel is an "invention", because that novel was not part of the world before being written, even if, of course, every possible novel exists as a possibility, because it's only a long string of characters.
-- "discovery" is the observation of something that is not evident at first sight, or that seems to be "surprising" or "not normal". This is of course a more problematic definition: something can be surprising for some people and normal for others. But in practice it's easy to agree on which facts of nature facts of nature could be qualified as "discoveries". Some examples:
- when Galileo Galilei observed that free falling bodies follow a law of squared times nobody expected a such simple regularity in nature.
- when you see Pythagoras' theorem without being told the demonstration, it seems a strange coincidence that the sum of two squares equals the other square
- when you see Maxwell's equations, it seems very strange that such simple symmetric equations describe so many facts of nature
- when you see the fundamental theorem of algebra, it seems to be a surprising coincidence that all polynomials of degree n have always n solutions
( I could continue with many more obvious examples, but I think I gave you the idea.. )

So, returning to the creation of mathematical worlds using axioms of our choice:
you can create a mountain of mathematical sentences made with casual axioms (a computer can do it even easily and very quickly), but it would be very improbable that something "interesting", or surprising would come out of them: at the same way as you can automatically create syntactically correct novels, or randomly built machines that are useful for nothing.
I think that axioms are like english grammar: they are the syntax of a language, but the meaning of the novel is something more than just a list of grammatically correct sentences.

But would this argument have the same force if applied to, say, wheels? Would we be surprised - or not - that aliens also have wheels? And would this mean that wheels (the quintessential 'invention') are therefore discovered? Do wheels have 'objective reality'?

Or imagine that aliens have cars - or at least, transport vehicles with four wheels. Would this mean that cars have objective reality? Or would it be that four points of contact with the ground works really nicely for stability? (more stable than 3, less unnecessary than 5 - are 4 wheeled cars an objective truth?) And that circular structures are good for things that move? Could it be that math is as it is for similar reasons? All of this is not to 'take a side' in the invented/discovered debate, but only to point out that the 'argument from aliens' is not a particularity strong one. At least, not without a whole bunch of other qualifications. — StreetlightX

The wheel is an invention related to the discovery of some physical facts of nature: the rolling friction of a round body is much smaller that the creeping friction of other shapes. Moreover, the center of a circle moves slower than the border. So, I can fix an axis to the center and use it to transport heavy objects with less force. Similar explanations can be given for most of the other technological inventions: they are related to the discovery of what we could call "laws of nature".
So, to answer your question, wheels (or cars) are a human invention (if we call "invention" the creation of an object that didn't exist before), but the laws of nature on which they are based are "an objective truth" that probably would be discovered by other intelligent beings and used to create machines similar to ours.

The question is, how far can we take that conclusion? When we develop mathematical theories and construct mathematical models to explain the regularities in our observations, do we thereby discover some objective truth about nature? — SophistiCat

I think the answer is YES, and I think there should be an objective way to distinguish if the regularities are due to the way we built our axioms or are objectively laws of nature

Interesting is however, that symmetry in a micro world, for example in the world of elementary particles, is exact, while in a macro world, for example in biology, is only approximate. Why is it so? — Hrvoje

I think the fundamental reason is that some properties of elementary particles (described by quantum mechanics) are intrinsically "discrete", whereas properties of objects in the macro world are continuous.
This is due to the fact that the only observable states of wave functions are the "eigenfunctions" of some symmetry operators. In other words we can only see states that have some exact geometric symmetries.
So, in a sense, the shapes of micro world are made of a limited number "perfect" mathematically defined forms.

The given proofs or the mathematical approach taken to establish a mathematical proof can be quite subjective and rely on the person doing the proof and what he or she has been interested in, yet mathematics as a whole is such a beautiful system that the truths aren't just inventions — ssu

.

I agree. But do you think is possible to give a concrete meaning (or measure) to what it means for a theorem to be "beautiful"?
I mean: if a large group of people is able to distinguish a beautiful mathematical theory from an ugly one, probably there exists a measure of "beautifulness" independently from the person that judges.

I agree with you when you say that

There is no a priori reason to favor those concepts over any other. So the reason will not be found in the abstract enterprise of mathematics, but in the world that we inhabit — SophistiCat

. So, I think that the concepts of numbers and geometrical objects are in some sense related to the physics of our universe, and not simply abstract logical constructions that can be included in a non contradictory theory.

You should be careful about how you ask this question. What mathematics are we talking about? Viewed from the most general perspective, mathematics is a logical game. You set up some rules and then you use those rules to construct abstract structures and prove theorems about them. There is an infinite variability of such games; the enterprise as such does not dictate to you which rules you should select and what you should construct from them, so there is nothing preexisting for you to discover. — SophistiCat

That's exactly my point: how can you decide if something abstract as for example a topological space corresponds to something physical (as in case of natural numbers) or is simply one of the infinite logical games that can be built out of our fantasy?
Well, my opinion is that not all logical games are capable of producing "interesting" results, and that "interesting" results are somehow independent of the logical rules and axioms that you use, but correspond to real characteristics of our universe.
So, if an intelligent culture completely independent from ours happens to create the same concepts out of the infinite quantity of possible logical games, that would be a strong indication that there is some meaning in these concepts that is not related to logical games.

Concepts like perfect circles, triangles and euclidean space are a description of some properties that a given class of physical objects have in common. At the same way, natural numbers describe the property of sets of physical objects of being put in a one-to-one correspondence.
So, an object is a triangle if it is made of three points connected by three straight lines. I can call a nail "point" and a piece of rope "straight line", if I know that i have to take into account only some of the properties of the physical objects. At the same way, I can call a set of five stones "5" and treat it as a number, if I use only the property of the stones to be distinguishable.
In this way, the definition triangle is "any physical object that can be recognized having three points connected by three straight lines", following an appropriately defined physical experiment.

But our maths axioms are not all based on reality (axiom of infinity for example) so I think certain parts of maths diverge from reality. — Devans99

From the point of view of mathematics, the only relevant thing is that the axioms that we invented are not inconsistent (i.e. not contradictory: they are satisfiable in some model). If the axiom of infinity is not inconsistent, there should be some model in which it is true; so in this model the axiom doesn't diverge from reality.
But you can prove the fundamental theorem of algebra even using a definition of complex and real numbers not even based on set theory and first order logic (for example type in homotopy type theory).
So, I would say that the theorem corresponds to some more fundamental fact of reality, even if the axiom of infinity could be only one of the many models that can be used to interpret the theorem.

It is interesting to note that according to relativity, euclidian geometry diverges from reality. But it is a useful approximation of reality and one that any aliens would no doubt have in their mathematical canon. — Devans99

Yes, so in my opinion euclidean geometry has an objective underlying reality, even if it doesn't correspond to the physical space-time.