Mathematics, Discovered or Invented?

[Mock Turtle]

The idea of independent discovery was correlated by the late Psychologist Carl Jung, with his theory of the collective unconscious. The collective unconscious would be analogous to brain firmware, behind human consciousness, that is common to all humans by virtue of our shared human DNA; genetically neural firmware evolving and defined over eons. These firmware or archetypes define us as a species; our collective human nature. Like the organs of the human body they are defined and refined over time on earlier foundations; archetypes.

Jung proved his thesis by showing many examples of very similar ideas appearing in many places with no proof there was any transmission of information. Our human mind's operating system are structured so similarly; eons of selective advantages, very similar conclusions or innovations would also be reached; consciousness will funnel to similar places. The steam engine was first invented in BC, but had little use until 18th century AD, where it appeared spontaneously, again. We have 6000 human language in earth, which would not needed or expected if there was only direct transmission, but no spontaneous innovation of similar types; language systems.

I tend to believe that math is a tool; fancy looking glass, that has many uses and has undergone refinement and lateral advancements; radio telescope. Mathematics is also like a faithful horse, that will go anywhere you lead it. We use math in computer games, to allow for endless lives. If you assume math is pure and natural, that means that this math not only predicts the after life but immediate reincarnation. I tend to believe that math is really more like a good horse, and will follow the lead of the human driver; will haul our foundation premises to market. All theories will use math to express them, even though most will eventually be replaced; different drivers of the same horse.

In the life sciences, statistical math is widely used. This is a very useful tool. However, the irrational bias, that math is pure and natural has led many to erroneously conclude, life is based on the horse and not the driver leading. It is quite bizarre, discouraging reason and common sense.

Division by a fraction is interesting in that allows us to violate conservation principles. If I have 1 kilowatt hour of electric energy and was to divide it by 0.5, I will get 2 kilowatts of energy. This violates energy conservation; energy cannot be created or destroyed, even if a most basic version of the math tool allows it on paper. This can occur when the driver has had too much to drink and the horse is leading. The faithful horse can know your routine and can even follow your unconscious lead.

In terms of life, life is very organized. The DNA is designed for specific template relationships. This allows life to replicate and duplicate itself with amazing accuracy over trillions of cells. So why use the statistical math horse, that is more geared to disorganized and unpredictable things; Lady Luck? It is like using a junkyard horse, who is used to pulling heavy scrap metal, to haul delicate produce. This will turn the perfectly ripe tomatoes and peaches, into bruised fruit, that appear less uniform at market. This is irrational for science; using the wrong math tool. If the horse is allowed to lead, the mind looks for flaws that are not there and then adds them; hammer leading the carpenter; we are all at risk.

My guess is this is about making money. There is more money in sickness and flaws than in genetic perfection. Although 1% may be sickness and flaws, this is 95% of the revenue, therefore everyone has to use the gold digger math and worship it as leading. The insurance company will also make money betting on accidents, which are a tiny fraction of all driving hours. We are taught to look at driving into terms of exception and that math horse, since this is where the money is lost and gained.

At least biology, now has a reason for its poor math horse choice. It should not take 10 years to come to market with a new drug that can save lives. This is due to the same math horsed used by Government Bureaucrats who think in terms of their career path and will stall until they are not liable or at risk. They do not think in terms of lives saved, but careers advanced against Lady Luck.

Didn't too well at maths then?

 

[Polymath257]

Wait ... as far as I know, except from very esoteric fields of maths, it is consistent, i.e. free of contradictions. It is incomplete, though, i.e. not every law can be proven or disproven.
I'm thinking about the Barber's Problem or the Tarsky Paradox. Those are examples of incompleteness, not of inconsistency.

Part of Godel's result is that it is impossible to prove that even basic (Peano) arithmetic is consistent without going to a more powerful system (which just opens up the issue of consistency of that system).

So, while it is true that we don't know of any contradictions from the Peano axioms (and I don't think anyone believes there are such contradictions), it is impossible to *prove* there are no contradictions. The same is true for *any* axiom system more powerful (that satisfies a basic condition that we can test any statement to see if it is an axiom).

 

[Polymath257]

Exactly. The basics of maths were found independently in Mesopotamia, India, China, South- and Mesoamerica at different times.

I think it might be relevant to say what is considered to be 'basic math'.

Pythagorus believed that all ratios should be 'rational': the ratio between two integers. It was proved this is not the case and the proof used the Pythagorean theorem.

What is interesting in the history of math is actually how dependent each culture was on transmission from other cultures. The spread of Greek mathematics because of the conquests of Alexander linked Greece, Mesopotamia, India, and influenced China. The notion of mathematical proof seems to have not arisen outside of that influence.

Another take on this is that some ideas (one, zero, simple addition) are ways that humans think. That does NOT mean it is 'discovery', but rather it is a form of species bias.

That is an advanced example.

I would disagree. Even you noted that calculus would be part of any basic system.

I think it, at least, hints at it. When something like the basics of maths are discovered multiple time - with no counter example - it is a strong hint that that is the only way to basic maths.

And yet, more modern investigations show that it is NOT the only way to do math. Even logic can be done in multiple ways (and has been in human culture).

Compare that to religion, the believers don't agree on the number or the attributes of the gods. We can pretty clearly assume gods to be invented, at least more so than maths.

 

[Polymath257]

Understood. And I concede all my other points. You've answered those objections very well, especially concerning gravity. And you've also shown that the Pythagorean theorem is not true in any kind of universal sense.

But I was wondering: are there axioms that are shared between both Euclidean and non-Euclidean geometry? (I honestly have no idea.)

Euclid had 5 postulates for geometry. What came to be known as the parallel postulate is the main difference between classical Euclidean geometry and the earliest versions of non-Euclidean geometry (Labochevskian geometry). But it is important to note that Euclid left out many axioms that would be strictly necessary for his results and modern investigations have looked at alternatives to Euclid's assumptions.

Because of the 'discovery' of non-Euclidean geometry, the direction of mathematics in the last 2 centuries has gone much deeper into the basic assumptions made prior to that point. Many 'axioms' that were previously considered to be 'intuitively obvious' are now seen as just more possible alternatives to a wide range of possible assumptions.

When most mathematicians talk about non-Euclidean geometry, they mean what follows from a collection of axioms that are 'close to' what Euclid assumed. But there are alternatives that go way beyond that.

If there aren't, then you can go ahead and ignore the rest of this post.

But if there ARE, then could a person (using JUST those axioms) form the "nub" of a (very limited) mathematical system? I'm being very hypothetical here. I'm not saying this geometric system would be very useful for anything... my question is: could such a system exist? And if it DID exist, would IT be discovered or invented. Hopefully that question makes sense. I admit, it's a weird question. Maybe even a silly question now that I read back over it.

It is interesting that Euclid did not make use of his 5th postulate for the first several results in his books. It is as if even he was uncertain about that axiom.

I'm working under the assumptions that the axioms themselves are absolutely discovered, ie. not invented at all. After all, if someone thinks the underlying axioms are invented, then of course all conclusions that follow would also be invented.

And that is where I am not so certain. My favorite analogy is with the game of chess. It is clear that someone invented the rules of chess: they are not something discovered.

But, given a specific configuration of pieces, it is possible to 'discover' that mate is possible in 5 moves.

Once the rules are selected, we can then discover the consequences of those rules. I think this is what happens in math: we invent our axiom systems (usually to correspond to some intuitions) and then discover the consequences of those assumptions.

Or maybe you could take it the other way and say that the non-Euclidean and Euclidean axioms all sort of "count." Euclidian talks about flat space. Non-Euclidean talks about curves. But, once one figures out if the space in question is flat or curved, the theorems that deal with that particular space are objectively true.

Your thoughts?

The issue is that we *define* flatness to be whatever Euclidean geometry studies. So this is a bit circular.

 

[Polymath257]

I would like you to comment on the Structuralist ontology of math if possible.
Structuralism, Mathematical | Internet Encyclopedia of Philosophy

Structuralism has some things in its favor. Benacerraf had a wonderful paper 'What Numbers Could not Be' that got structuralism going and he had some very good points. But I still think the ontology is problematic.

I keep going back to the example of chess. In what sense does the game of chess exist? In what sense does a specific game of chess exist?

It is clear that the game of chess does NOT depend on any particular physical realization. It does NOT require small pieces colored black or white and a physical checkered board to move upon. It is certainly possible to use a computer generated board and 'pieces' that are no more than certain patterns of electron motion. But we can go further and note that even that isn't actually necessary: it is possible for two people to 'play a game of chess' completely in their minds simply by communicating the moves they make.

In this sense, there is no ontology of chess. The game is *purely* structural. There are no 'objects' in chess that are necessary for the game. There is *only* the 'office' (to use your link).

And I would say the same about mathematics. We select the axioms and rules of deduction (which correspond to the rules of chess) and attempt to deduce results from those axioms (corresponding to playing a game of chess). As long as we follow the rules of deduction, we have played a valid game of math.

The primary difference is that we choose the rules of math so that we can model certain aspects of the real world. We want to 'count' and so we invent axioms that allow us to do that. And, we find that the axioms we choose give us a good model of certain aspects of the world around us. But this is discovered by testing in a wide variety of situations. We discover when those abstract axioms are useful and when they are not.

As an example, we have an axiom system that says that 4+5=9. The corresponding test in the real world might be to put 5 stones in a bag and then put 4 more into the bag. We then look in the bag and count how many stones are in it. if we count 9 stones, we say that the rules of arithmetic work in that case.

But, if we put 5 stones in the bag and then 4 more into the bag and we then go to count the stones and find 20 stones, what do we say? We say that some stones 'broke up' or that maybe there were stones in the bag prior to our activity. We find reasons why 5+4=9 is not applicable to the situation. We have tested the assumption and found a case where the conclusion fails and then *go to other models*. That particular model has been falsified in that case.

What we have found is that there are 'conservation laws' that we can form mathematically and test, finding that they work in the real world. In the above case, we have a 'conservation of number of stones' in some cases, but not in others.

Anyway, my position about mathematics is primarily formalism: we have certain axioms and rules of deduction. Mathematical proof consists of playing by the assumed rules.

 

[TagliatelliMonster]

One of the age-old questions that we haven't debated recently (at least not in the last year).

I think that mathematics is discovered. One piece of evidence is that maths has been discovered multiple times independently.
We assume it even to be so universal that aliens on other planets must have discovered it, if they have technology.

Tagging @Polymath257

I'm going with a mixture of both, depending on the angle we are looking at it.

On the one hand, I'ld say that math, as a language, is invented.
However, the things that this invented languages describes and models, are discovered.

And sometimes, we discover new things / relations for which we need to invent new math to be able to model it. Like when Newton came up with calculus.

With math it is perfectly possible to model things that simply do not exist in reality or which don't match reality.
Just like with regular language like english, we can form sentences or arguments which are also wrong eventhough they are internally consistent.


To draw an analogy...
We didn't invent the sun.
We discovered the sun and invented language to describe that discovery.

 

[Ostronomos]

Structuralism has some things in its favor. Benacerraf had a wonderful paper 'What Numbers Could not Be' that got structuralism going and he had some very good points. But I still think the ontology is problematic.

I keep going back to the example of chess. In what sense does the game of chess exist? In what sense does a specific game of chess exist?

It is clear that the game of chess does NOT depend on any particular physical realization. It does NOT require small pieces colored black or white and a physical checkered board to move upon. It is certainly possible to use a computer generated board and 'pieces' that are no more than certain patterns of electron motion. But we can go further and note that even that isn't actually necessary: it is possible for two people to 'play a game of chess' completely in their minds simply by communicating the moves they make.

In this sense, there is no ontology of chess. The game is *purely* structural. There are no 'objects' in chess that are necessary for the game. There is *only* the 'office' (to use your link).

And I would say the same about mathematics. We select the axioms and rules of deduction (which correspond to the rules of chess) and attempt to deduce results from those axioms (corresponding to playing a game of chess). As long as we follow the rules of deduction, we have played a valid game of math.

The primary difference is that we choose the rules of math so that we can model certain aspects of the real world. We want to 'count' and so we invent axioms that allow us to do that. And, we find that the axioms we choose give us a good model of certain aspects of the world around us. But this is discovered by testing in a wide variety of situations. We discover when those abstract axioms are useful and when they are not.

Your post is somewhat muddled.

All languages are said to possess structure. Therefore, they literally exist as objects, processes, interpretations and reality. It is strikingly obvious to even a casual observer. However, this fact has remained elusive for many centuries. Due in part to poor understanding.

L:M->R means that language maps mind to reality.

As an example, we have an axiom system that says that 4+5=9. The corresponding test in the real world might be to put 5 stones in a bag and then put 4 more into the bag. We then look in the bag and count how many stones are in it. if we count 9 stones, we say that the rules of arithmetic work in that case.

But, if we put 5 stones in the bag and then 4 more into the bag and we then go to count the stones and find 20 stones, what do we say? We say that some stones 'broke up' or that maybe there were stones in the bag prior to our activity. We find reasons why 5+4=9 is not applicable to the situation. We have tested the assumption and found a case where the conclusion fails and then *go to other models*. That particular model has been falsified in that case.

You are grasping at straws here. The math is sound. It is your interpretation of it that is at fault.

What we have found is that there are 'conservation laws' that we can form mathematically and test, finding that they work in the real world. In the above case, we have a 'conservation of number of stones' in some cases, but not in others.

Anyway, my position about mathematics is primarily formalism: we have certain axioms and rules of deduction. Mathematical proof consists of playing by the assumed rules.

Are you saying that our perspectives parallel those of mathematics only under certain conditions? Because I would go further and say all conditions.

 

[sayak83]

Structuralism has some things in its favor. Benacerraf had a wonderful paper 'What Numbers Could not Be' that got structuralism going and he had some very good points. But I still think the ontology is problematic.

I keep going back to the example of chess. In what sense does the game of chess exist? In what sense does a specific game of chess exist?

It is clear that the game of chess does NOT depend on any particular physical realization. It does NOT require small pieces colored black or white and a physical checkered board to move upon. It is certainly possible to use a computer generated board and 'pieces' that are no more than certain patterns of electron motion. But we can go further and note that even that isn't actually necessary: it is possible for two people to 'play a game of chess' completely in their minds simply by communicating the moves they make.

In this sense, there is no ontology of chess. The game is *purely* structural. There are no 'objects' in chess that are necessary for the game. There is *only* the 'office' (to use your link).

And I would say the same about mathematics. We select the axioms and rules of deduction (which correspond to the rules of chess) and attempt to deduce results from those axioms (corresponding to playing a game of chess). As long as we follow the rules of deduction, we have played a valid game of math.

The primary difference is that we choose the rules of math so that we can model certain aspects of the real world. We want to 'count' and so we invent axioms that allow us to do that. And, we find that the axioms we choose give us a good model of certain aspects of the world around us. But this is discovered by testing in a wide variety of situations. We discover when those abstract axioms are useful and when they are not.

As an example, we have an axiom system that says that 4+5=9. The corresponding test in the real world might be to put 5 stones in a bag and then put 4 more into the bag. We then look in the bag and count how many stones are in it. if we count 9 stones, we say that the rules of arithmetic work in that case.

But, if we put 5 stones in the bag and then 4 more into the bag and we then go to count the stones and find 20 stones, what do we say? We say that some stones 'broke up' or that maybe there were stones in the bag prior to our activity. We find reasons why 5+4=9 is not applicable to the situation. We have tested the assumption and found a case where the conclusion fails and then *go to other models*. That particular model has been falsified in that case.

What we have found is that there are 'conservation laws' that we can form mathematically and test, finding that they work in the real world. In the above case, we have a 'conservation of number of stones' in some cases, but not in others.

Anyway, my position about mathematics is primarily formalism: we have certain axioms and rules of deduction. Mathematical proof consists of playing by the assumed rules.

It seems that what you prefer is a form of mathematical fictionalism. Mathematics is a make believe game with mathematical entities and relationships invented fictions. This however, creates the problem of why this game cannot be played any which way and why the game is so useful or generalizable. And if mathematics is a fictional game, then discovery of new mathematical knowledge is not possible, which seems clearly not to be the case. There are other problems
Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)

 

[Polymath257]

It seems that what you prefer is a form of mathematical fictionalism. Mathematics is a make believe game with mathematical entities and relationships invented fictions. This however, creates the problem of why this game cannot be played any which way and why the game is so useful or generalizable. And if mathematics is a fictional game, then discovery of new mathematical knowledge is not possible, which seems clearly not to be the case. There are other problems
Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)

Yes, we can choose any axiom system we want and play the game with that system. But we choose axiom systems that have the flexibility to make models of reality. Those systems are useful precisely because they have that flexibility and the models we make can be tested and pass the tests.

We gain knowledge in exactly the same way a chess player gains knowledge about chess. We explore different games and see what we can do. Knowledge about math is knowledge about what follows in some axiom system.

For example, the continuum hypothesis is known to be independent of the usual axioms of set theory. We can choose to adopt it or choose to adopt its negation. Either option is equally valid. At this point, neither alternative seems to give more interesting math or any useful models, so there is no consensus about which way should be adopted.

I don’t think the continuum hypothesis has a truth value.

 

[Polymath257]

Your post is somewhat muddled.

All languages are said to possess structure. Therefore, they literally exist as objects, processes, interpretations and reality. It is strikingly obvious to even a casual observer. However, this fact has remained elusive for many centuries. Due in part to poor understanding.

And I see that as simply false. Languages are not objects.

L:M->R means that language maps mind to reality.

You are grasping at straws here. The math is sound. It is your interpretation of it that is at fault.

And that is exactly what it means for the model to fail. All we have shown is that the law of 'conservation of number of stones' is false in some situations. That model is an *interpretation* of the mathematical formalism and can be tested to see when it works in the real world.

Are you saying that our perspectives parallel those of mathematics only under certain conditions? Because I would go further and say all conditions.

Well, for example, 2+2=4 fails if you add 2 quarts of water and 2 quarts of alcohol. You will NOT get 4 quarts of the mixture. So that additive model for volumes fails in this case.

We make mathematical models to help us understand the world around use. We choose basic axioms to allow for the construction of those models. But we then need to go and *test* those models to see if they actually work in practice. if not, we try to make a different mathematical models. Those that work in a wide variety of cases are ultimately called 'physical laws'.

 

[Alien826]

I suppose every culture will eventually have a need to count stuff or measure stuff. It may be somewhat innate, at least in terms of determining size and amounts. I can imagine people being aware of who has bigger piles of food or which animals are bigger. I'm not sure about animal species, although I've heard about some species where the mother is able to count their offspring (though I'm not sure if that's true or just an old tale).

I saw an experiment where a dog was shown a number (say 5) objects, treats to get its attention, which were then put behind something. When only four were placed there the dog continued to search for the fifth treat. That didn't happen when all five were placed.

 

[Brickjectivity]

No, it says something about right triangles *if Euclidean geometry is assumed*. But, since the early 1800's, we know that there are non-Euclidean geometries that are just as internally consistent as Euclidean geometry. In these geometries, such basic 'facts' as Pythagorus' theorem and that the sum of the angles of a triangle is a straight angle are simply false. Furthermore, such geometries are more appropriate in many situations (for example, spherical geometry is non-Euclidean as is the geometry of general relativity).

I can easily imagine situations where an alien race would first arrive at a non-Euclidean geometry where Pythagorus' theorem fails.

Its tangential, but in non Euclidean (which I have just started looking into) differential geometry, you have hyperbolic and elliptical, right? The hyperbolic is analogous to (but not the same as) a Cartesian system that has been wrapped onto a saddle shape while the elliptical is analogous to a Cartesian that has been wrapped onto a pill sort of shape. The analogy has some commonalities with the results of hyperbolic or elliptical geometry in that a triangle in hyperbolic geometry has less than 360 degrees in it internal angles, and a triangle in elliptical geometry has more than 360 degrees internally.

Again its tangential, but its interesting. You don't have to belabor if you are busy with more relevant points and other posts, but if you feel like it I'm curious what sort of aliens. Would these be aliens that were shapeless like octopi perhaps?

 

[sayak83]

Yes, we can choose any axiom system we want and play the game with that system. But we choose axiom systems that have the flexibility to make models of reality. Those systems are useful precisely because they have that flexibility and the models we make can be tested and pass the tests.

We gain knowledge in exactly the same way a chess player gains knowledge about chess. We explore different games and see what we can do. Knowledge about math is knowledge about what follows in some axiom system.

For example, the continuum hypothesis is known to be independent of the usual axioms of set theory. We can choose to adopt it or choose to adopt an alternative. Either option is equally valid. At this point, neither alternative seems to give more interesting math or any useful models, so there is no consensus about which way should be adopted.

I don’t think the continuum hypothesis has a truth value.

The question is why are fictional games useful at and so fundamental to the physical reality?

 

[Polymath257]

The question is why are fictional games useful at and so fundamental to the physical reality?

They have enough flexibility to become a type of language. They aren't fundamental to reality. They are fundamental to the way we understand reality.

Similarly, the English language is very useful for understanding reality, but is not fundamental to reality.

Limited mathematical structures can be studied (and are), but they don't generally allow for the range of testable models we like when studying science. On the other hand, the vast majority of math has very little impact on models: it is simply done for pleasure by mathematicians. if the resulting language is found useful later, so be it.

 

[sayak83]

They have enough flexibility to become a type of language. They aren't fundamental to reality. They are fundamental to the way we understand reality.

Similarly, the English language is very useful for understanding reality, but is not fundamental to reality.

Limited mathematical structures can be studied (and are), but they don't generally allow for the range of testable models we like when studying science. On the other hand, the vast majority of math has very little impact on models: it is simply done for pleasure by mathematicians. if the resulting language is found useful later, so be it.

If math is not fundamental to reality, then reality could be known as accurately without mathematical systems. Can you show how this can be done?
I think we have discussed about relational quantum mechanics which interpret quantum mechanics so that reality is fundamentally not about objects with stable properties but rather emerge out of relationships that arise during interactions (traditionally called observations). If this view is correct and if, as per Structuralist, mathematics is the study of all possible self consistent abstract structures that exist....then we have an explanation as to why reality requires mathematics for us to know it. This is why I prefer to the structuralism approach.

 

[Alien826]

Division by a fraction is interesting in that allows us to violate conservation principles. If I have 1 kilowatt hour of electric energy and was to divide it by 0.5, I will get 2 kilowatts of energy.

Don't you actually get two half-kilowatts? That's the question that division answers. How many 0.5s are in 1? Answer 2. Not two 1s, two 0.5s.

 

[Alien826]

Similarly, the English language is very useful for understanding reality, but is not fundamental to reality.

We only have to read a few threads here to see the problem with this! I'm not being totally serious, but my observation is that language more often confuses understanding than advances it. That's why any serious scientific discussion tends to involve very specific agreed upon meanings of the words used.

On the main subject, I tend to think "invented" rather than "discovered". That's because pretty much all (actually all?) mathematics involves axioms that cannot exist in the physical world. Examples might be a perfectly flat surface, a perfect circle, and objects that are exactly equal to one another.

 

[Polymath257]

If math is not fundamental to reality, then reality could be known as accurately without mathematical systems. Can you show how this can be done?

math is a language. We could also use English, but it is not nearly as compact.

I think we have discussed about relational quantum mechanics which interpret quantum mechanics so that reality is fundamentally not about objects with stable properties but rather emerge out of relationships that arise during interactions (traditionally called observations). If this view is correct and if, as per Structuralist, mathematics is the study of all possible self consistent abstract structures that exist....then we have an explanation as to why reality requires mathematics for us to know it. This is why I prefer to the structuralism approach.

Understood. And I like the general approach of structuralism, but dislike the Platonic overtones. It all comes down to that word 'exists'. I am hesitant to use that word in mathematics *except* as a technical term saying we can prove a statement that has a backwards E.

 

[Polymath257]

If math is not fundamental to reality, then reality could be known as accurately without mathematical systems. Can you show how this can be done?
I think we have discussed about relational quantum mechanics which interpret quantum mechanics so that reality is fundamentally not about objects with stable properties but rather emerge out of relationships that arise during interactions (traditionally called observations). If this view is correct and if, as per Structuralist, mathematics is the study of all possible self consistent abstract structures that exist....then we have an explanation as to why reality requires mathematics for us to know it. This is why I prefer to the structuralism approach.

A couple of comments.

1. Mathematics can, and does, look at 'inconsistent structures'. Paraconsistent logic allows some statements and their negations to both be true. However, unlike standard logic, a contradiction doesn't prove every statement (and thereby trivialize the structure).

2. I'm not so sure the structuralist and the formalist approach are that different. I look at axiom systems and what can be shown from them. The point is that every axiom system is a 'structure' and all definable structures are axiom systems. So I think the main difference is one of ontology. Do those 'structures' actually 'exists' as structures? Or are they actually just collections of assumptions and conclusions with no other ontology?

Again, the game of chess is a structure. To what extent and how does it 'exist'? I ultimately see the game of chess as a (rather simple) axiom system which doesn't allow for much model building.

 

[sayak83]

math is a language. We could also use English, but it is not nearly as compact.

Understood. And I like the general approach of structuralism, but dislike the Platonic overtones. It all comes down to that word 'exists'. I am hesitant to use that word in mathematics *except* as a technical term saying we can prove a statement that has a backwards E.

I do not think writing stuff in English will do the trick. Sure you can describe by words what the math symbols are doing. But it is still referring to the same mathematical operations only. For example what would be the non-mathematical way to describe the eigenvalue problem of quantum wavefunction relationships from which Energy or Momentum states are derived for a physical system?
As I see it, if it's a fictional game, it seems to be a necessary fiction to understand reality....which does not look like a property of fictional entities (like Superman of Santa Clause) or games (like chess or football).