I think the question is poorly posed because it does not take into the account the extent to which mathematics is grounded in human sensory-motor schemata. I think that mathematics can be said to be "objective" to the degree to which the sensory-motor schemata from which it is ultimately abstracted can be said to be objective. That is, I think that mathematics definitely has some purchase on reality, although it's also easy to see that this purchase is always "approximate", much in the same that way that the contents of sensation and cognition in general are also approximate.

I do not think that mathematical entities exist independently of the mind. That said, I don't doubt that the relational structures embedded in mathematical theories mirror the vastly more complex relational structures existing "out there" in mind-independent reality. Regarding aliens, it would not surprise me in the least if an intelligent alien culture with a significantly divergent anatomy and physiology developed a mathematics that is all but incomprehensible to us (and vice versa).

Do wheels have 'objective reality'? — StreetlightX

The form of a wheel has objective utility. Mathematics is an elaboration of counting, and counting also has an objective utility. It is also an objective fact that things are countable. It seems reasonable to presume that any alien mathematics would necessarily, if it is to qualify as mathematics at all, be based on counting.

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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

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 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.

You're basically saying that the difference between invention and discovery is that the former comes into existence while the latter always existed. I agree with this.

What I find relevant to your question is that nothing precludes a correspondence/match between invention and discovery. 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).

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.

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

The first alien world I visit the population has the same symbolism for math, develop the same proofs, and apply the math the same way. I think to myself there must be some underlying objective reality for this math. I move on to my second alien world and find this population has the same symbolisms and proofs but do not use math at all. There is no application of math in science, business, or sport. I ask someone why is this so, and they reply because these symbols and rules are very beautiful and that is enough for us. I begin to wonder about my belief about underlying objective realities. The last planet I visit again has the same symbols and proofs but there is a problem. Most of time the answers they get for math problems is sometimes right and sometimes wrong. There seems to be no consensus on which is right. Additionally, there application of math is sometimes successful and sometimes not. I begin to think to myself that maybe all this talk about underlying objective realities is irrelevant, and what is more interesting is how alien species handle symbols in their worlds.

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". — Mephist

Are you confining your argument to mathematics alone or are you also including the mathematical nature of the laws of nature?

There's a difference and it matters.

If you're only talking about math then your argument seems a bit weak because it's not impossible that many or all intelligent life construct identical or similar mathematical worlds. I know many examples where people thought they hit upon a "new" idea, only to discover that it was an old one. Coincidences do occur. Of course, using mathematical probability, we can deduce that parallel development of identical/similar mathematical worlds among different intelligent life is suspicious to say the least. Nevertheless, remember we share the same universe and are exposed to the same elements that stimulate quantitative (math) thinking. It shouldn't be a surprise that we created identical/similar math. In fact it is to be expected.

On the other hand if you say that the laws of nature are universal and mathematical then the argument is stronger since it implies that our universe has a mathematical structure and so we must discover it and not invent. Of course we have to first invent the math and then check if it applies to the world outside. I'd like to refer you to Eugene Wigner's 1960 article The unreasonable effectiveness of math in the natural sciences

If you took a shot for every time someone in this thread unnecessarily qualified something with 'objective' (objective truth, objective fact, objective reality objective utility(?)), you'd straight up die. Skip drunk and go straight to alcohol poisioning induced death.

I do not think that mathematical entities exist independently of the mind. That said, I don't doubt that the relational structures embedded in mathematical theories mirror the vastly more complex relational structures existing "out there" in mind-independent reality. — Aaron R

Scientific method relies on the ability to capture just those attributes of subjects in such a way as to be able to make quantitative predictions about them. In other words, if you can represent something mathematically, then you can use mathematics to make predictions about it. The greater the amenability of the subject to mathematical description, the more accurate the prediction can be: hence the description of physics as the paradigm of an 'exact science'. Bertrand Russell said that 'physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.' But nevertheless, within that domain, the ability to apply mathematical logic to all manner of real objects yields practically all of the power of scientific method. In other words, what can be expressed in quantitative terms can also be made subject to mathematical analysis and, so, to prediction and control. It becomes computable. That is of the essence of the so-called 'universal science' envisaged on the basis of Cartesian algebraic geometry.

But this also challenges the dichotomy of mathematics being 'in the mind' and the world 'out there'. Things that can be quantified to conform to mathematical predictions in the same way that they conform to logic. We know by mathematics and logic the laws and axioms which are visible to thought itself - Frege's 'laws of thought' - and requiring no empirical validation, on account of them being logically necessary; they're not 'out there'. But through this quantitative method, the certainty availed by logical prediction can be applied to collections of all kinds, not just to empirical particulars , and with mathematical certainty. It's the universal applicability of these procedures to practically any subject which opens access to domains of possibility forever out of reach of an intelligence incapable of counting (whence, Wigner's 'unreasonable efficacy').

I think that in some fundamental respect, real numbers are discovered rather than being invented, but that they're neither in the mind, nor in the world; they pertain to a logical domain which transcends that dichotomy; they are real a priori to that division; but that having the ability to perceive them also bestows the ability to invent other like kinds of ideas; so the distinction 'discovered' or 'invented' can't be hard and fast.

Bertrand Russell said that 'physics is mathematical not because we know so much about the physical world, but because we know so little — Wayfarer

:cheer: :up: Maslow's hammer

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.

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. — Mephist

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.

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

I would argue that a lot of things that we take as important yet problematic are indeed mathematical, but simply not computable. Even the patternless are still mathematical.

You seem to be in favor of the view that math is a discovery rather than an invention. I think that's just too strong a conclusion i.e. unsupported by the evidence.

What is the evidence?

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). That the laws of nature can be described with numbers and geometry indicates the universe has a mathematical structure. This is as real,thus discovered, as math can ever be.

I think you disagree with what I said above. You made an argument in your last post about the film nirvana and how a person in a simulation could still discover math despite the laws of nature in that simulation being an illusion. Doesn't your argument actually refute your position which, I think, is that math is a discovery? Without a connection to the real world math would only exist in our minds; making it an invention.

Personally speaking, I think math is both an invention and a discovery. I don't think math is a complete discovery because there are many mathematical objects that have no real world counterparts i.e. they exist only in the mind. However, some mathematical objects have real world application i.e. an actual phenomenon matches a given mathematical object.

Let M ={all math objects}
P ={phenomena in the real world}

The intersection of M and P is not the null set. However M - P too is not the null set. Math is both an invention and a discovery.

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.

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)

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.

Hi. I think this might be relevant. One of the frameworks of reality is time. A temporal context seems to be a necessary aspect of the laws of nature. However, whether time is real and discovered or unreal and invented is yet to be resolved. Math could be a coincidence.

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.

So my idea is that the axiomatization of mathematical ideas is invented, but our axiomatizations are based on some underlying objective facts of nature that are discovered. And the distinction of what is real from what is invented could be based on a definition of this kind (is it possible to make this definition more precise?) — Mephist

No, I think it's already just a little too precise. I think it should be something like this:

The axiomatization of mathematical ideas is invented, but our axiomatizations are based on some many underlying objective facts observations of nature that are discovered.

The axiomatization of mathematical ideas is invented, but our axiomatizations are based on some many underlying objective facts observations of nature that are discovered — Pattern-chaser

I believe it's not only from observations of nature that mathematics takes inspiration.
An obvious example: Mandelbrot set is not present in nature, but it's an interesting object. Group theory has started from the solution of polynomial equations, and only after was discovered to be important in physics. Maybe one of the most surprising is Riemanian geometry: invented as a pure abstraction, and then discovered to be exactly what's needed to describe gravitation. Often is the other way around: mathematicians discover some interesting structures that you can build out of pure axiomatic theories, and then it comes out that they are exactly what's needed for physics.

Yes, some bits of maths are interesting, and most of it is useful too. But this topic asks whether maths is invented or discovered. To believe that maths was discovered is to mistake the map for the territory. [ It also avoids the rather obvious criticism that there is nowhere in the real universe that we can point to and say "there's maths" or "Oh, look, there's a sine wave". ] For maths is a mapping tool. It helps us understand the universe, in some ways, and that is admirable. But it wasn't discovered, it was invented, just like real, literal maps made of printed paper. There's no shame in that. :smile:

Well, my idea is that there is a "map" that exists in some kind of Platonic world that is taken as a model by nature. Parts of this map are taken as a blueprint from nature to build real things and even human beings. The real things are not perfect as the map, but tend to resemble to them in a high degree. Maybe there are even parts of nature that don't follow any map, but the ones that follow the map are the ones that we may be able to understand.

Maybe there are even parts of nature that don't follow any map, but the ones that follow the map are the ones that we may be able to understand. — Mephist

Isn't everything itself a perfect model of itself?

Following a map is somewhat difficult metaphor to understand. You see if there is a pattern, then we can extrapolate from the pattern. Yet something can be patternless, which still has a perfect model of itself and that is itself.

In my opinion we don't call mathematics what is patternless. It's not clear for me what a pattern is, but I would argue that nobody would consider a mathematical paper that express patternless theorems. And I think laws of nature for some reason prefer mathematical models to patternless ones.

In my opinion we don't call mathematics what is patternless. — Mephist

That's the whole point! It's genuinely defining the limits of computable math. What here is important is to understand just how basic patterns are for ordinary mathematics.

I would argue that nobody would consider a mathematical paper that express patternless theorems. — Mephist

Think so?

When we don't have a pattern, we can't extrapolate, calculate or do the other usual mathematical stuff. Yet of course something not having a pattern is still logical and still part of mathematics. Same thing with immeasurability or non-measurability. Take for example the non-measurable sets like the Vitali set.

When we don't have a pattern, we can't extrapolate, calculate or do the other usual mathematical stuff. Yet of course something not having a pattern is still logical and still part of mathematics — ssu

Yes, what I meant is that some parts of mathematics are "interesting" and some are not. And I think this distinction can be made internally to mathematics itself, without looking at nature ( see my other topic https://thephilosophyforum.com/discussion/5789/is-it-possible-to-define-a-measure-how-interesting-is-a-theorem ). So theorems, or rather theories, that have a high value of the "interest" function ( the ones that have a pattern ) are discovered, because the value of the "interest" function for that theory is defined for all theories even if we don't know them. On the other hand, 10 + 10 = 20 is of course logical and true, but there's nothing in it that makes it "interesting"

Same thing with immeasurability or non-measurability. Take for example the non-measurable sets like the Vitali set — ssu

I believe non measurable sets are interesting as part of topology, but they are not a model for the physical space. For the physical space we should use a model where all functions are continuous, so that the Banach-Tarski theorem is false and all objects have a non-zero measure (but possibly an infinitesimal one), such as for example in (https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis), or any other simpler axiomatization that has these characteristics. If the axiomatizations are not equivalent, which one of them is correct is a matter of physics, not mathematics.

But this topic asks whether maths is invented or discovered. To believe that maths was discovered is to mistake the map for the territory. For maths is a mapping tool. It helps us understand the universe, in some ways, and that is admirable. But it wasn't discovered, it was invented, just like real, literal maps made of printed paper. There's no shame in that. :smile: — Pattern-chaser

Well, my idea is that there is a "map" that exists in some kind of Platonic world that is taken as a model by nature. Parts of this map are taken as a blueprint from nature to build real things and even human beings. The real things are not perfect as the map, but tend to resemble to them in a high degree. Maybe there are even parts of nature that don't follow any map, but the ones that follow the map are the ones that we may be able to understand. — Mephist

I cannot help but admire your ambition. :smile: You have renamed our map as a plan, something made beforehand to describe what will be made, instead of something created later, to help navigate. So now you have reversed the roles of the map and the territory, suggesting that we actually have the plan and the territory. [ I.e. where the plan is the master/reference, and the territory is a secondary copy. ]

And so, regrettably, I feel compelled to ask the difficult question: where is this "Platonic world", where the plans for the Universe are stored until they are needed? For if the map/plan exists, and this Platonic world is where it exists, then where is this Platonic world? Your surmise seems to rest upon your having an answer to this question, doesn't it? :wink: :chin: