Is Maths an underlying principle of nature or did it originate from our brain?

[Lorentz]

Is Maths an underlying principle of nature or did it originate from our brain?

I think we would agree that Maths is abstract, but would it be there if we wouldn't be here?

Maybe Maths is an underlying principle of nature that our brain is capable of taking notice of in an abstract way. ?

Or is Maths just an inevitability of nature?

What's your view about this?

 

[Zurtex]

There's a question, would:

$$e^{\pi i} + 1 = 0$$

if we weren't here? I think that's a question about reality not really about maths...

 

[Lorentz]

I think that's a question about reality not really about maths...

It wasn't meant as a question. It was meant as a think over.

 

[Lorentz]

There's a question, would:

$$e^{\pi i} + 1 = 0$$

if we weren't here?

A marsian would eventually derive the same equation...

 

[rdt2]

Are natural laws invented or discovered? Does the tree make a sound if it falls in the forest and there's no-one there?

These are interesting but I guess Zurtex's point is that they might be better discussed in the Metaphysics forum.

Artoo.

 

[Lorentz]

Are natural laws invented or discovered? Does the tree make a sound if it falls in the forest and there's no-one there?

These are interesting but I guess Zurtex's point is that they might be better discussed in the Metaphysics forum.

Artoo.

I thought Mathematicians would have an opinion about this. I know the Pythagoreans had.

And besides classifications only exist in the human brain. There is no exact place to put it. It's just a question of choice. I didn't post to get into an argument about where I should have put this.

 

[matt grime]

Speaking up for (some of ) the mathematicians: we don't care. If we did we'd be doing philosophy. Would the martians have derived that equation? Perhaps, perhaps not - they almost certianly wouldn't have devised the same way of presenting it, and we couldn't tell if they'd picked i or -i as their square root of -1, which they may have called something else anyway. That answer has a superficial and a non-superficial part to it.

 

[Lorentz]

Speaking up for (some of ) the mathematicians: we don't care. If we did we'd be doing philosophy.

The old Greek society would turn around in their graves!

It's a philosophical question, but that doesn't mean it doesn't affect Maths. I think it's quite ignorent studying Maths without questioning it's origins.

 

[bureado]

Yes! It's a philosophy subject, but scientists also need an "out-of-the-box" reflection to know why they're here or doing what they do.

I think maths would be there if we wouldn't, and that's because I think maths are not just the derivation of an expression or any kind of tangible reflection about a problem. I think maths are fundamental and exact laws that manage lots of processes in any existent system in the Universe, and even if we couldn't understand them (or don't want to), they would always be there.

Finally, I think that religion and beliefs would definitely _not_ be there if humanity were familiar with maths since prehistory.

Greetings.

 

[Lorentz]

Yes! It's a philosophy subject, but scientists also need an "out-of-the-box" reflection to know why they're here or doing what they do.

I think maths would be there if we wouldn't, and that's because I think maths are not just the derivation of an expression or any kind of tangible reflection about a problem. I think maths are fundamental and exact laws that manage lots of processes in any existent system in the Universe, and even if we couldn't understand them (or don't want to), they would always be there.

Finally, I think that religion and beliefs would definitely _not_ be there if humanity were familiar with maths since prehistory.

Greetings.

Thanks!

 

[matt grime]

It's a philosophical question, but that doesn't mean it doesn't affect Maths. I think it's quite ignorent studying Maths without questioning it's origins.

The truth or otherwise (if such a thing can even be said to be true) of your query does not impinge upon our ability to do maths. It is thus of little consideration to mathematicians. It might be of interest to a mathematical philosopher (Wittgenstein's Philosophical investigations c. entries 200 or so, or Russell and Whitehead's proof that 1+1=2).

Personally I think it's dangerous to make sweeping statements about the ignorance of other people (and mathematicians in particular) when you might not know all the facts (should we mention your misspelling of 'ignorant' as well as the mistake with the apostrophe in 'it's'?). I speak as a mathematician (one paid to do mathematical research), and reflect the opinions of most of the colleagues I know, certainly including a Fields Medal winner who said of something similar 'that is a question we can safely leave to the philosophers'. If it's any consolation I think Connes might disagree with me, and he's astronomically cleverer than you or me.

Questioning its origins in anycase is not the same as worrying about its metaphysical properties.

 

[ahrkron]

I think to some extent this depends on what you call "math". Is math "the laws that manage processes", as bureado said? I really don't think so. Math does not "manage" things. It is our way of describing them. What governs natural phenomena is a set of basic interactions (which sounds much more like physics to me). Macroscopically, there are consequences of this interactions that we have measured and that we have found how to link to the fundamental components. Math is basically the way in which we code this links.

Without any intelligent race in the universe, fundamental interactions would still add up, in the same way, to macroscopic phenomena, but there would be no one to summarize rotations in $$i$$ plus exponentiation, or to invent various descriptive frameworks for related phenomena, and to later feel awe when finding relations between those frameworks.

 

[Janitor]

One of the philosophical camps likes to say something--and I've come across it on about three different occasions, so they have promulgated it effectively--like: If there was a dinosaur in a pond, and another dinosaur came along and walked into the pond, then there were two dinosaurs in the pond.

Obviously their position is that math is an aspect of the physical world, and doesn't need a human brain, or in fact any biological structure, to delve into it for it to be real.

 

[HallsofIvy]

There is a "philosophy" section to Physics forum and this probably belongs there.

However, there are some mathematicians that are very interested in the philosophy of mathematics!

It has been said (and if I were really good, I'd know by whom immediately) that "God created the positive integers- all the rest is the work of man."

Actually, I would disagree with that- the abilility to talk about "1" and "2" requires the ability to distinguish between objects- that's easy for, say, elephants, but what about elementary particles? Or, for that matter, slime mold? I remember seeing an article in the newsletter of the Society for Industrial and Applied Mathematics in which the author boldly stated that a spiral galaxy is a clear example of a spiral occurring in nature! Good thing it wasn't on PBS or I would have chucked a rock through the screen! (I'm hard on televisions.)
The "spiral" one sees in a galaxy (or the seeds in a sunflower) is a far cry from the "spiral" defined in mathematics.

Mathematics consists of variety of "models" to which we can, to better or worse approximation, associate various physics situations (not to mention economics and other studies that have nothing to do with "nature"). I suspect that mathematics tends to model the way we think about things rather than any natural properties of the things themselves. But we have so many different mathematical models that may not be a useful way of thinking about it- We may just model every possible way to think about things.

 

[Janitor]

Leopold Kronecker was the mathematician who said that about integers. If I remember, he and Cantor were always at loggerheads about their viewpoints.

I remember getting excited to read, as a teenager, about integers showing up as quantum numbers for various bound states in quantum mechanics, such as the simple harmonic oscillator. But then it finally sank in that the analysis of phenomena in nature invariably involves making simplifying assumptions, and potential fields in the real world are not "perfect" in their properties, so real-world particles cannot be expected to behave exactly in such a way as to have energy levels that are related as ratios of whole numbers, or what have you.

QUESTION ADDED LATER: Halls, I see in your profile that you are at Gallaudet University. I am old enough to remember a Fallujah-style uprising by students there. Okay, so I exaggerate. If I recall, they wanted some/all(?) of the teachers/administrators(?) there to be deaf. Were you around back then?

 

[Zurtex]

I am actually one who is interested in the philosophy of maths. Something I was trying to ask was:

$$e^{\pi i} + 1 = 0$$

Fundamental to the universe or is it simply a construction of our own minds? Do we need apply our own sense of reality to the universe to make maths work? It seems we certainly do for the physics created at anyone point...

 

[quddusaliquddus]

Maybe it's fundamental to the universe, and so we have constructed it in our minds.

 

[Janitor]

Or, to give another example in addition to Zurtex's elegant one, should we really expect some alien race out there in space might be working contentedly with a plane geometry in which the cube of the right-triangle hypotenuse length is equal to the sum of the cubes of the side lengths? Given how the square aspect of the Pythagorean relationship is connected to so many things in math and physics, they couldn't design machines--including spacecraft --very successfully if they were going with cubes!

 

[motai]

Maybe it's fundamental to the universe, and so we have constructed it in our minds.

I agree, the human brain has a tendency to recognize patterns out of chaos.

 

[jdavel]

It seems to me (although I'm no mathematician) that a new branch of math could be invented (with definitions and axioms). But from that point on, it's all discovery. For example, Newton is usually credited with the "invention" (or co-invention) of calculus. And caclulus does seem like an invention, like a tool. On the other hand, the Taylor series seems like a discovery, once the rules of calculus were there. Which gets to the question about e^(i*pi). Without the Taylor series, I don't think anyone would have ever realized that it equaled -1 (unless there's another way to show it). So it's a discovery within a mathematical regime that was invented as tool for making discoveries in the physical world. How's that?

 

[HallsofIvy]

It seems to me (although I'm no mathematician) that a new branch of math could be invented (with definitions and axioms). But from that point on, it's all discovery. For example, Newton is usually credited with the "invention" (or co-invention) of calculus. And caclulus does seem like an invention, like a tool. On the other hand, the Taylor series seems like a discovery, once the rules of calculus were there. Which gets to the question about e^(i*pi). Without the Taylor series, I don't think anyone would have ever realized that it equaled -1 (unless there's another way to show it). So it's a discovery within a mathematical regime that was invented as tool for making discoveries in the physical world. How's that?

That's actually a very sharp and well-accepted point. Are "theorems" discovered or invented? Both! Theorems or, more generally, mathematical statements are invented when someone establishes the basic axioms of the mathematical structure. They are then discovered when they are proven.

 

[Nexus[Free-DC]]

I have always felt that the whole structure of mathematics inhabits a sort of shadowy world that is halfway between the real world and imagination. It's sort of like a hotel vacancy. You can never see or feel, weigh or measure a hotel vacancy but nobody would seriously argue that it is just a figment of some concierge's imagination.

And some areas of maths are more 'real' - that is more applicable to the real world - than others. I find arithmetic, group theory, calculus and geometry to really describe how various things in our universe behave. Whereas some of the more abstract areas of maths like function theory and topology, with their contrived and inelegant proofs, just seem like a lot of fanciful jiggery-pokery.

 

[matt grime]

I would argue that topological proofs are far more elegant than calculus ones (proof that Pi_2 is abelian as opposed to the horrendous nature of some calculus proofs). And the current vogue is to use higher dimensional algebra - ie categories usually of topological objects - to model (topological) qunatum field theory.

For those who think that theorems are discovered: how do you explain the continuum hypothesis? If it's true in some physical sense, then it's true...

 

[Nexus[Free-DC]]

That's a good point. There are a fair few ugly proofs in any discipline of maths and frequently one is able to switch between disciplines to get a nicer answer. I know a couple of calculus proofs that are tedious the normal way but when you express them in matrix or vector form the result falls out almost immediately.

But what I meant was many of the proofs in the more abstract areas of maths make use of bizarre constructions that seem to me to be artificial and contrived. Whereas ugly proofs in other areas tend to consist of a lot of messy working.

I guess I am a bit biased against Topology because I couldn't understand my lecturer's heavily accented english or read his handwriting. Still, I humbly suggest that anyone who thinks the proof of the Mayer-Vietoris theorem is elegant ought to give me some of whatever they are smoking.

 

[Jez]

Is Maths an underlying principle of nature or did it originate from our brain?

My Professor said the other day that humans didn't invent math, we just discovered it. Everything from calculas to the golden ratio has been embedded into nature and it will still be there even if we wern't here to see it.

 

[Lorentz]

My Professor said the other day that humans didn't invent math, we just discovered it. Everything from calculas to the golden ratio has been embedded into nature and it will still be there even if we wern't here to see it.

But isn't an invention like that as well?

An invention is also embedded into nature. Nature gives us a potential of making certain things. When we invent the television we realized the potential of nature of producing a television. The invention of the television has always been possible within nature. E.g. if we can't make a time machine it means nature doesn't have the potential of a time machine.

So an invention isn't that different to a discovery, or is it?

 

[Lorentz]

If we now compare the television with Maths...

The television works on principles of nature (physical laws). Without them we couldn't have invented the television. We regard those physical laws as discovered. With those physical laws the television isn't an so obvious object to make even though it always has been possible to make it. We as we call it, invented the television. It's something we made with help of nature. But it's not what nature thought of. I mean nature didn't think: Let's make these laws so humans can build a television.

Now Maths is build upon physical laws or how ever you would call them (Just look at the dinosaur example). Upon these laws we build a whole network which we call Maths. Nature didn't have Maths in mind when creating nature (or maybe it did), but it was inevitable a possibility with the physical laws given to us by nature. So Maths has always been there within nature... part of it being physical law and part of it being an inevitability of those laws (without nature thinking of those inevitabilities). So part of it being discovered and part of it being invented (but we already arrived at that conclusion, I think).

Does this make any sense to you?

 

[matt grime]

Still, I humbly suggest that anyone who thinks the proof of the Mayer-Vietoris theorem is elegant ought to give me some of whatever they are smoking.

Topology is a younger subject than analysis. The proofs in older subjects are tidied up over the years as we realize better what the 'essence' is (your own example shows that idea). Compared to some proofs Mayer-Vietoris is quite straight forward, but perhaps in 30 years some one will have a better proof for it. (I think there is already using gluing functors and such.)

 

[Zurtex]

My Professor said the other day that humans didn't invent math, we just discovered it. Everything from calculas to the golden ratio has been embedded into nature and it will still be there even if we wern't here to see it.

It's a nice thought and that is partly to do with why I choose to go into maths rather than physics. I would like to believe that maths is pure truth, that long after we are gone and the universe looks very different that:

$$\frac{d}{dx} \left( e^x \right) = e^x$$

But it may just be an abstract construction of our minds as a pragmatical jump to predicting what goes on around. I mean can you really prove that $1 - 1 = 0$ or do we just define it to be so for ease of use?

 

[matt grime]

But part of the reason why this question belongs in philosophy is that even though that professor believes these things have some existence independent of ours, and I don't, is that if you were to see us solve a problem you couldn't tell which of us had which view. Ask your professor, if he thinks somethings will be mathematically true even if we weren't here, how he treats the continuum hypothesis. Or for that matter the idea that the natural numbers are a set.