The Question : is mathematics discovered or invented?

[Pythagorean]

Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshïta at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we ‘got something out’ of the equations that we did not put in.”

The approximations that come into QM at this point make my head spin.

I remember feelng like I got hit with a five-car approximation train, BAM BAM BAM BAM BAM! one lecture very recently

 

[CaptainQuasar]

In contrast to Pythagorean's intelligent and insightful comment, http://www.penny-arcade.com/comic/2003/12/19" that reminded me of this thread. (The shouting, gesticulating guy would be me, of course.)

⚛​

 

[oldman]

The article by Wigner, so kindly unearthed by Zz, pretty much says it all:

Re WHAT IS MATHEMATICS?, Wigner says: "mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts"

Re WHAT IS PHYSICS? he says: "The physicist is interested in discovering the laws of inanimate nature." (my emphasis).

From sunny, warm SA: I'm off for a few weeks to what I think is the lovliest city there is: Cape Town. So I'm signing off now. Thanks for the company, folks. Great discussion!

Oldman.

 

[CaptainQuasar]

Perhaps it demonstrates conceit on my part but if I'm not afraid to disagree with Penrose I'm not afraid to disagree with Wigner (who both, by the way, are physicists, right?)

From sunny, warm SA: I'm off for a few weeks to what I think is the lovliest city there is: Cape Town. So I'm signing off now. Thanks for the company, folks. Great discussion!

Happy trails!

⚛​

 

[krimianl99]

The labels and belief sets related to it are invented, the relations are discovered.

Thus it is mostly discovered in my opinion. It is really just a subset of deductive reasoning anyways to which the same question and answer applies.

 

[salguodojibwa]

mathematics is law

what does life or the universe care of your ponderings. Mathematics is a universal law. It is its own master and was neither discovered nor invented. It sought us out as it has all sentients.

 

[a2tha3]

there is no need for thumbs. We didn't invent them, we're just lucky enough to have them.
in my analogy though, thumbs are discovered, it is the inches that were invented...

This is very interesting to say that the inches were invented, because If I were to apply what you just said to math, but instead of your thumb use numbers (just lucky enough to have them) and then use equations, (which were invented according to your post;consider the equations the inches) then you could come to the conclusion that mathematics was invented.

I have been skimming through this topic, and found some excellent points for both sides.. I wonder would it be fair to say that Some of mathematics was discovered, while other parts of it were invented?

I mean, does it have to 100% invented or 100% discovered?

If you get into geometry, and think of shapes..were those invented? A simple shape in the beginning of time.. (Im sure you can think of some)

Actually I think I can argue everything is invented, because no one has really set restrictions on the word "invented".. does a volcano invent lava?

Forgive if I am not making much sense, I am having a hard time thinking with this dazzling headache, and hope most of you can get the gist of what I am trying to say.

 

[Pythagorean]

I mean, does it have to 100% invented or 100% discovered?

I've been thinking on this

A metaphor would be that if the somebody invents a plane, they can still make discoveries about the plane; discoveries that they did not invent, but are a consequence of their invention.

Discovering something about an invention is obviously discovery, so I can't refute that, but I'm not sure if the mathematics itself is being discovered or discoveries are being made about the mathematics that was invented (i.e. we discover that the mathematics is consistent if we follow the rules... we may discover that two rules we invented imply (or force) a third rule... but was that third rule purely discovered? I think it was invented when the first two rules were invented, and then discovered, but not purely discovered... I guess that's the point I'm getting at...)

In the end I guess there's the hidden question in this topic of whether mathematics exists in nature independent of humans... and I think it does not... I think it's a consequence of the way we think, how we like to organize things in our minds, how much we appreciate things "making sense".

As pointed out in the article that ZapperZ posted, we're lucky that math works out for us in regards to the physical sciences... well, some would call it luck. I call it brute force (a lot of people have been working at this for a lot of centuries... we were bound to get somewhere)

 

[CaptainQuasar]

BTW a related thread I've come across is [THREAD=201057]Tegmark's Mathematical Universe[/THREAD]. Not that I would consider this “Mathematical Universe Hypothesis” to be quite the same thing I'm saying, but it seems like a product of thinking along the same lines about whether mathematics is invented or discovered.

⚛​

 

[CaptainQuasar]

I'm willing to bet I could hear the definition a hundred different ways and still have some ambiguity left.

Just noticed this from last month: I'd point out that this is true of any definition whatsoever, not just the definition of mathematics. Human languages simply aren't the sort of thing where all ambiguity can be removed.

Along those lines I would mention that the word “discover” is pretty ambiguous in it's own right, e.g. “Christopher Columbus discovered America.” I think that one would have to get more specific about the meanings of the words “discovered” and “invented” before seriously tackling this question.

As I said above, there are of course all sorts of completely trivial ways in which mathematics was invented: mathematics has words in it and words are invented, mathematics uses chalkboards and chalkboards are invented, some constructs in mathematics are clearly just devices that are used to teach mathematics to humans, some constructs in mathematics are intermediate devices to assist humans in understanding more complex mathematical concepts, etc.

Basically my view in this is that considering anything that is present in every single darn field of study - like words, physical paraphernalia, teaching constructs, or intermediate conceptual devices and frameworks - within the scope of inquiry for whether mathematics is discovered or invented, is pointless and not really addressing the question. To put it another way, as I said above, I think the real question is whether the subject of study of mathematics is discovered or invented.

⚛​

 

[krimianl99]

what does life or the universe care of your ponderings. Mathematics is a universal law. It is its own master and was neither discovered nor invented. It sought us out as it has all sentients.

Yeah... Has to be one or the other lol...

Anyways the natural laws and our ideas related to them are two different things. An important distinction to make especially when dealing with concepts like infinity.

You can remove ambiguity by using a special type of definition that refers to the context a word is supposed to be describing. Then you just use deductive reasoning and experiences to determine the properties of that context.

 

[CaptainQuasar]

You can remove ambiguity by using a special type of definition that refers to the context a word is supposed to be describing. Then you just use deductive reasoning and experiences to determine the properties of that context.

Ah, well, go ahead and remove the ambiguity from [post=1621309]my definition of mathematics[/post] for Pythagorean, would ya? Thanks.

⚛​

 

[Pythagorean]

To put it another way, as I said above, I think the real question is whether the subject of study of mathematics is discovered or invented.

⚛​

This is where I am too and my point (to a certain degree) is there is no 'subject of study' outside of your mind. Our brains have evolved (for some reason or another) to be able to think in a way that's mathematic-like. This is not special alone (there are animals that can 'count'). What is special about it to us is that we have self-awareness (or at least the illusion of it) and we can philosophize about the technique we use naturally, and eventually come to an agreement (remember, this is purely a human agreement about mathematics... and furthermore it's an agreement between a VERY SMALL fraction of the human race). These agreements are... (axioms perhaps). Only humans care or think about these axioms. The way beasts use 'mathematics' is the way you run from a swarm of bees as opposed to running from a bee... or would rush out of the way of a water tower dumping it's contents on you, but wouldn't be so frightened by the rain.

Mathematics rose out of human contemplations of interaction with the physical world. How we perceive the physical world is not perfect... we are limited to our sensory and processing. As a result of our supposed self-awareness, we are able to use our imagination to predict things that we can't sense... and it is from this that math was born... and pushed and poked and prodded to fit into place with the physical world. In fact, no mathematical formula will ever be able to predict a real-world event with 100% accuracy (down to the location of every elementary particle at given time...)

Time and space do not flow 1,2,3,4 like we imagine... order is a severely unstable thing in the real world... We're too huge (and too tiny) to notice the complete aspect of reality. Mathematics is a sort of human utopia, an Elysium Fields in this aspect that helps us to imagine beyond our senses.

 

[kenewbie]

I want to propose a neat little view I have on this:

Until such time as we have a working theory of everything that can be proved (which also requires a working way to prove something), *everything* is inventions. e v e r y t h i n g.

We are making stuff up to describe the what we see, until such time where we discover the truth.

k

 

[CaptainQuasar]

This is where I am too and my point (to a certain degree) is there is no 'subject of study' outside of your mind.

Then are you saying that you do not believe mathematics nor things isomorphic to it would be duplicated by a distant alien species, because they do not have human minds?

Mathematics rose out of human contemplations of interaction with the physical world.

That sounds like a physicist's description of mathematics. As a computer engineer I'm inclined to disagree and an accountant for example might disagree as well.

Modern computers obey Turing and von Neumann's principles not because those guys had modern computers to study and extrapolate from and not because modern computers were built to obey those principles, but because mathematical concepts like Turing machines and the various things they're isomorphic to are describing something that exists external to the human mind, some congruence in reality that is fundamental enough to be apparent both in the physical world and in mental/logical function and analysis in general.

In fact, no mathematical formula will ever be able to predict a real-world event with 100% accuracy (down to the location of every elementary particle at given time...)

Time and space do not flow 1,2,3,4 like we imagine... order is a severely unstable thing in the real world... We're too huge (and too tiny) to notice the complete aspect of reality. Mathematics is a sort of human utopia, an Elysium Fields in this aspect that helps us to imagine beyond our senses.

I again think you're talking about physics here. Physics is the discipline that concerns itself with building perfect models of the universe, then falls continuously short in its Sisyphean effort. This is almost like one field of study making Freudian projections on another.

⚛​

 

[Pythagorean]

This is almost like one field of study making Freudian projections on another.

⚛​

this made me smile

Then are you saying that you do not believe mathematics nor things isomorphic to it would be duplicated by a distant alien species, because they do not have human minds?

no, I think this is irrelevant.

The point is that without self-consciousness, mathematics does not exist. Aliens can be defined as hive-like or human like (up to the imagination so far) but either way my fundamental view remains the same. Mathematics is a product of self-consciousness.

I again think you're talking about physics here. Physics is the discipline that concerns itself with building perfect models of the universe, then falls continuously short in its Sisyphean effort.

Absolutely, I am... things like accounting follow human rules... of course things will work out perfect there... it's and invention designed to work with a previous invention (mathematics) for human comfort and ease!

All perfect models in mathematics are without doubt inventions (abstractions of human thinking). The only thing that can be discovered is the physical world outside of our brains. If there was no physical world, there would be no reason for a consciousness to create mathematics.

 

[CaptainQuasar]

It seems to me that we may have looped back to a trivial sense of "invented". If every product or analytic activity of self-consciousness is invented then everything is invented, whether or not the invention is prompted by something completely external to all conscious minds.

⚛​

 

[Pythagorean]

It seems to me that we may have looped back to a trivial sense of "invented". If every product or analytic activity of self-consciousness is invented then everything is invented, whether or not the invention is prompted by something completely external to all conscious minds.

⚛​

Not exactly. We all adhere to the same physical reality, everybody doesn't get to invent their own physics... but scientists (like Newton) do invent the math (or symbolism/diagrams) to help them understand the physical world. We're pattern making creatures, it's absolutely in our nature. We invented supernatural things to explain phenomena so that the pattern could be fulfilled. Math is a more sophisticated version of this need for us to complete patterns to make sense of things.

 

[meaw]

Is Maths correct ?

Hi,
This is my first posting. I wonder while we are spending so much time and energy in redefining either QM or GR, has anyone tried redefining Maths?
Here is why I think we should take a fresh look at Maths.

1. Neton said - space, time, mass and all other fundamental quanties are absolute. some derived quantities like velocity are relative. Laws of nature are absolute.

2. Einstein said - Only Laws of nature are absolute, space-time-mass everything is relative

3. GR said, space-time is not only relative, they freaking bend too !

4. I say - not even maths is absolute, it should vary too!

I have objection to Platonic view of absolute mathmatical reality. Why should mathematics be absolute, just because it makes life easier to assume so?

what if 1+1 is not 2 always. Under heavy curvature of spacetimemass, 1+1 may be 1.5.
In other word let the number line be bent :) any problem ?

Does anyone know if there are other who may have been thinking like this idiot ?
I consulted the doc, he suggested I should do some Yoga !

 

[oldman]

Not exactly. We all adhere to the same physical reality, everybody doesn't get to invent their own physics... but scientists (like Newton) do invent the math (or symbolism/diagrams) to help them understand the physical world. We're pattern making creatures, it's absolutely in our nature. We invented supernatural things to explain phenomena so that the pattern could be fulfilled. Math is a more sophisticated version of this need for us to complete patterns to make sense of things.

Since it has seemed such a struggle for folk to decide whether maths is invented or discovered, or indeed what maths IS, it might be worthwhile to go back to beginnings and consider where maths came from, and what kind of animals develop and use it. I've now suggested this in another thread (What is maths?).

I fully agree with what you say here. We are driven to make and recognise patterns and, I'd add, to talk about and describe them almost ad infinitum. Witness this thread!

 

[oldman]

...Penrose ...Wigner (who both, by the way, are physicists, right?)

No. Not Penrose. He's a mathematician who writes superbly about speculative or mysterious parts of physics. But his work on tiling (Penrose tiling) did have a big impact on solid state physics when Schectman unexpectedly discovered (quasi) crystals with previously-thought-impossible five-fold symmetries, long ago.

Your later remark "Physics is the discipline that concerns itself with building perfect models of the universe, then falls continuously short in its Sisyphean effort" is lovely.

But this would be theoretical physics since about 1975. Not the physics that wins Nobel prizes e.g. for discovering the phenomenon of giant magnetoresistance that makes storing all this talk stuff compactly on disc so easy.

 

[Noone]

sorry but math was here before we were so the answer is no to both... it just wasnt called math nor did people know how to show it to others like how we use words. the world and the means of how it works and the ways to measure it were allways here or there, its just our thought is not. It was man the defines and lables all things for the means of communication to others. math was also difined for the same thing, to give understanding of how, what, and why the workings of the world work how they work. but then if you see it in a difrent way it would be yes to both... mainly philosophy is chaotic because the all most infinit ways of perception of the point of views of one kind of thought -.- but even now if all the math was never written down we could look to nature to unravel it again... just like how they did it back in the day of stick's and stone's :D

 

[Noone]

first you must discover the concept to invent somthing of that concept... so its like the question of which came frist the chicken or the egg... we all know one thing, and that's the chicken came from the egg... so the invention came from the discovery of the concept. its the answer is discoverd :D doesn't matter much were the egg came from maybe chickens didnt always lay eggs x.X silly question this one is. answer is logical. its discoverd. you must discover your invention be for you can invent somthing :P

 

[Noone]

are unknown start is the egg were the chickens. we know were we came from :D just don't know were the egg did

 

[Noone]

this question has been de-bunked by logical and reasonable thinking :P

 

[Noone]

Reasonable thought is what brings control over a chaotic philosophy, logic binds it by telling what must be true and what must be false.

 

[Noone]

so for us to invent somthing you must discover your inventions concept. aka the foundation for your invention to be made or said or shown

 

[CaptainQuasar]

Not exactly. We all adhere to the same physical reality, everybody doesn't get to invent their own physics... but scientists (like Newton) do invent the math (or symbolism/diagrams) to help them understand the physical world. We're pattern making creatures, it's absolutely in our nature. We invented supernatural things to explain phenomena so that the pattern could be fulfilled. Math is a more sophisticated version of this need for us to complete patterns to make sense of things.

There have been quite a few different and conflicting versions of physics!

And if you're talking about symbolism and diagrams, that's again something I've pointed out is common to all disciplines. The symbolism and diagrams of physics are just as invented.

Phenomena in the physical world were already constrained by the calculus Newton and Leibniz each independently discovered while trying to model it. They couldn't have simply invented any mathematics they desired to use to model it. And I would think there were conclusions they were forced into by the nature of the mathematics that they did not anticipate via physical observations - conclusions arrived at by examination of the congruences within the math itself, not through experimental inquiry.

You can make up any story you want to for explaining supernatural phenomena - witches, ghosts, UFOs, psychic abilities, etc. and there need not be any congruence or particular structure within the explanations. Not so with mathematics.

And as I pointed out, it's the history of physics which has been much more like a flawed theorizing about mysterious phenomena and retelling of a story, rather than mathematics. On the contrary, it's the mathematics which have been consistent and unchanging throughout human history - Euclid's axioms and your namesake formula have held against all scrutiny down through the ages and are still employed in modern mathematics. Try that with Aristotle's explanation of gravity or of the fundamental elements that make up all matter (ie. Earth, Air, Fire, Water, Aether to the ancient Greeks).

So, I hate to say it because it sounds imputing, but it still looks like you're projecting. Physics is the discipline where someone can make up whatever they want and if they're deft enough in speech and mathematical legerdemain and authoritative flair can get away with it. But in mathematics, if you try to simply make something up that isn't true, your symbols and diagrams prove themselves to be false all on their own. And false conclusions will break other parts of mathematics (often quite obviously) if any attempt is made to use them. In mathematics there is some external constraint that is more immediate and forceful than the experimental confirmation that science is limited to.

⚛​

 

[Pythagorean]

There have been quite a few different and conflicting versions of physics!

And if you're talking about symbolism and diagrams, that's again something I've pointed out is common to all disciplines. The symbolism and diagrams of physics are just as invented.

Phenomena in the physical world were already constrained by the calculus Newton and Leibniz each independently discovered while trying to model it. They couldn't have simply invented any mathematics they desired to use to model it. And I would think there were conclusions they were forced into by the nature of the mathematics that they did not anticipate via physical observations - conclusions arrived at by examination of the congruences within the math itself, not through experimental inquiry.

You can make up any story you want to for explaining supernatural phenomena - witches, ghosts, UFOs, psychic abilities, etc. and there need not be any congruence or particular structure within the explanations. Not so with mathematics.

And as I pointed out, it's the history of physics which has been much more like a flawed theorizing about mysterious phenomena and retelling of a story, rather than mathematics. On the contrary, it's the mathematics which have been consistent and unchanging throughout human history - Euclid's axioms and your namesake formula have held against all scrutiny down through the ages and are still employed in modern mathematics. Try that with Aristotle's explanation of gravity or of the fundamental elements that make up all matter (ie. Earth, Air, Fire, Water, Aether to the ancient Greeks).

So, I hate to say it because it sounds imputing, but it still looks like you're projecting. Physics is the discipline where someone can make up whatever they want and if they're deft enough in speech and mathematical legerdemain and authoritative flair can get away with it. But in mathematics, if you try to simply make something up that isn't true, your symbols and diagrams prove themselves to be false all on their own. And false conclusions will break other parts of mathematics (often quite obviously) if any attempt is made to use them. In mathematics there is some external constraint that is more immediate and forceful than the experimental confirmation that science is limited to.

⚛​

I wasn't really talking about physics, I was talking about our physical reality. Of course physics (the discipline) is an invention, but the physical relationships are what is discovered. In response to most of the rest of your post, I'll point this out again:

(keep in mind that the author also brings up your points, but neither him or I see the conflict that you purport)
ON MATH:

Somebody once said that philosophy is the misuse of a terminology which was invented just for this purpose. In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms.

The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

ON PHYSICS:

The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

MATH IN PHYSICS:

Naturally, we do use mathematics in everyday physics to evaluate the results of the laws of nature, to apply the conditional statements to the particular conditions which happen to prevail or happen to interest us. In order that this be possible, the laws of nature must already be formulated in mathematical language. However, the role of evaluating the consequences of already established theories is not the most important role of mathematics in physics. Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool.

A possible explanation of the physicist's use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

all from http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

In other words... the fact that we fail so much using math in physics should be a big hint that mathematics is not some shining grail that holds the universe together. The universe would still be here if math wasn't, we just wouldn't understand it the way we do. Math is a perfect world, the universe is not. It's absolutely spontaneous, random, and diverse.

A punch in the nose is much more real than the laplacian. However, we can both agree that the laplacian is more useful.

 

[Pythagorean]

also, here's a post I found that was made independent of this discussion here at PF:

https://www.physicsforums.com/showthread.php?t=221532#7