Roger Penrose – Is Mathematics Invented or Discovered? [video] • Hacker News Board

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

#151

Post by sorokod » Fri, Apr 17, 2020, 2:22 PM UTC

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 precision.

Those initial axioms, were they discovered or invented?

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

#152

Post by cossatot » Fri, Apr 17, 2020, 2:22 PM UTC

phaemon wrote: post #110

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…

And yet, most blind mathematicians are geometers, which suggests your theory is incorrect.

We could perhaps modify the theory, thinking of the mechanics of research. Progress is made by an individual in the niches where others have not managed yet. Therefore the advantage in geometry for a blind mathematician may be in a different mental model of spaces.

It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]

recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]

This is a quote from a book by Alexei Sossinsky, quoted here: https://onionesquereality.wordpress.com/2011/07/31/blind-geo...

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

#153

Post by Trasmatta » Fri, Apr 17, 2020, 2:24 PM UTC

RashadSaleh wrote: post #140

ebj73 wrote: post #132

A vaguely, but tangentially related fact, is that Stephen King actually considers the stories in his books as discovered, rather than invented. He talks about it in his book 'On Writing'. He considers the stories to be sort of preexisting things, and his job as a writer to unearth and discover them, rather than invent them. It's about his frame of mind while writing, I guess, but still.

I doubt that he actually said that but I could be wrong.

This is actually a big plot point in The Dark Tower books.

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

#154

Post by StandardFuture » Fri, Apr 17, 2020, 2:27 PM UTC

henrik_w wrote: post #136

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…

I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode) https://corecursive.com/035-bartosz-milewski-category-theory...

And really one of the biggest pieces of proof of this is that some of the earliest mathematics developed was planar geometry.

But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world.

This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.

But, this makes sense because we really don't ask the same question about human language. We almost never ask: is human language constructed or discovered?

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

#155

Post by wincent » Fri, Apr 17, 2020, 2:28 PM UTC

ebj73 wrote: post #132

A vaguely, but tangentially related fact, is that Stephen King actually considers the stories in his books as discovered, rather than invented. He talks about it in his book 'On Writing'. He considers the stories to be sort of preexisting things, and his job as a writer to unearth and discover them, rather than invent them. It's about his frame of mind while writing, I guess, but still.

To me this sounds like an echo of the view that free will is an illusion. The stories that King is going to write are already predetermined by the laws of physics applied to the initial conditions of the system. In a sense, he's really just "along for the ride".

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

#156

Post by Tainnor » Fri, Apr 17, 2020, 2:29 PM UTC

otabdeveloper4 wrote: post #143

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…

> Math ... describes the world around us to the utmost precision. It does no such thing. The models we create using mathematical language do. Math is a precise language that can be used to describe the world around us precisely. It can also be used to describe utter nonsense or utter fantasy precisely.

Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc. Although at least to some extent statistics is the one branch of mathematics that does seem to be applicable (but the way this influences results, as opposed to methods, seems to be mathematically not that interesting).

There have been a lot of efforts of modelling social phenomena, the arts, etc. mathematically and while there are some interesting partial results (e.g. that languages can to some reasonable extent be analysed by parse trees or some aspects of music), most "grander" theories I have seen do not really stand up to much scrutiny.

I'm not an expert, but I could assume that part of this is due to a lot of nonlinearity in the phenomena studies, which means that many classical methods don't work well; maybe chaos theory etc. could shed more light on these things, but I don't know enough about it.

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

#157

Post by Noughmad » Fri, Apr 17, 2020, 2:31 PM UTC

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 find that if you take this set of axioms, then this statement is true. Is that an invention or discovery? I would call it a discovery, although different from a discovery of a natural phenomenon.

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

#158

Post by naringas » Fri, Apr 17, 2020, 2:34 PM UTC

A bit both:

e.g. The circle is an invention. But the number pi (it's value) is discovered

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

#159

Post by david_w » Fri, Apr 17, 2020, 2:38 PM UTC

henrik_w wrote: post #136

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…

I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode) https://corecursive.com/035-bartosz-milewski-category-theory...

This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge.

It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.

The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality is mediated by our brain's own idiosyncrasies and limitations of thinking.

It's no more surprising that external objective reality is mathematically predictable and describable than it is that our eyes can see shapes and some, but not all, light. That's the purpose of eyes and they tell us enough of what we need to know that we can survive.

Our brains are exactly the same thing- they tell us a story in a way which helps us to survive. Actual reality may be beyond our capacity to conceive of or worse, may seem like nonsense to us because it's aggressively illogical or specifically contradictory and reality therefore makes no "sense" to us.

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

#160

Post by analbumcover » Fri, Apr 17, 2020, 2:39 PM UTC

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 much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.