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Roger Penrose – Is Mathematics Invented or Discovered? [video]

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Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#161

Post by magicalhippo »

analbumcover wrote: post #160
magicalhippo wrote: post #76

If B follows from A via logic, but A is invented, isn't B invented as well then? Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered. However to me the basic premise, take some objects which aren't numbers but which square to a n…

It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be…

Well if one would consider numbers and their product "discovered", then my point was that introducing the objects which have the property that they are not numbers but square to numbers seems to me like an invention.

At least if there is to be any meaningful distinction between a discovery and an invention.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#162

Post by roywiggins »

sorokod wrote: post #151
DeathArrow wrote: post #24

I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe th…

to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy". Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precis…

Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.

Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#163

Post by ableal »

Kaiyou wrote: post #80
hliyan wrote: post #67

Earlier quoted context omitted.

I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of ge…

Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.

> on this planet

Location was not specified ...

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#164

Post by mmmBacon »

danck wrote: post #49

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

This is an excellent point. When I took algebra as an undergraduate I was blown away by the fact that you can choose any axioms and then derive an algebra based on those axioms. I was blown away because prior to that course I just assumed that our “standard” axioms were immutable.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#165

Post by roywiggins »

Noughmad wrote: post #157
scrozart wrote: post #148

Phenomena are discovered. Mathematics are invented to describe them.

Not all mathematics though. Math works by starting with a set of axioms. These are chosen, usually but not necessarily such that they will describe some kind of physical reality. That is the part which you invent to describe phenomena, like the Peano axioms which describe quantities of countable things. But then, you discover (and of course also prove) things that follow from these axioms. So a mathematician would fi…

Geometry and arithmetic and calculus all existed before axioms and worked okay. You can do quite a lot of mathematics "naively". It's only in the 20th century that axiomatization became really important, because naive set theory turned out not to work very well.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#166

Post by roywiggins »

deltron3030 wrote: post #96

Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.

You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results.

The world of ZFC doesn't feel very constrained by the actual universe, is all.

And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#167

Post by foldr »

Kaiyou wrote: post #80
hliyan wrote: post #67

Earlier quoted context omitted.

I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of ge…

Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.

That's only true if you limit this to conscious thought. Lots of organisms perform impressively complex calculations. The desert ant odometer is one classic example:

https://jeb.biologists.org/content/210/2/198

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#168

Post by naasking »

DeathArrow wrote: post #2

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics t…

> Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0]

The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#169

Post by captain_clam »

justforfunhere wrote: post #38

Could the same be said about just about everything? Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error. Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discov…

Think of it this way: Anyone can invent a new song, and no matter how derivative or lacking in artistic merit, they can still claim to have composed a new piece of music.(La-la-la-laaa. There, I just did it! Short but sweet.)

But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."

Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]

#170

Post by tasuki »

itvision wrote: post #97

There's one thing I cannot wrap my head around. Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of infl…

> If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.

It can also be liberating and uplifting. You're a part of the universe - you are the universe expressing itself. The result of uncountable generations of brutal evolution, you're quite a miracle!

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