My take on this is that Mathematics is such a vast area of investigation - much larger than Nature itself - that there is in fact a mix of both. In that regard it is close to Engineering, where in an attempt to design something on might stumble upon things or perform some types of research and discovery, and vice versa.
Roger Penrose – Is Mathematics Invented or Discovered? [video]
271–280 of 322 posts
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#272Earlier quoted context omitted.
Tell me more about the difference between "potential infinity" and "actual infinity". Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity? I've been thinking a lot about infinity this past week, mostly because of Conway's lect…
The video timestamp you linked does not say "free will exists". Is says "humans has free will implies element particles have free will".
His definition of free will is quite limited. He only suggests that free will is limited to a "free" choice between options within one's power.
Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#273Earlier quoted context omitted.
> it's an observation that there is no sharp line between invention and discovery This is true of pretty much any human categories, yes. That does not mean the categories aren't valid or that they should all be collapsed into one.
Fair enough. But notwithstanding, I contend that my claim is not vacuous. I would say that this particular failure of categorization actually reveals a deep truth, namely, that computation is universal.
> I would say that this particular failure of categorization actually reveals a deep truth, namely, that computation is universal.
I agree that computation is universal, but I don't see drawing a distinction between discovery and invention as denying that fact. In "computation is universal" language, the distinction is just the observation that there are different kinds of computations--moreover, there are different kinds of computations in the category "human mental processes". We don't need to say they're all the same to recognize that they're all computations at bottom.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#274This is a silly semantics game and a boring one at that. the presumption that the words are contradictory, or that mathematics is a monolithic entity that plays a single role of invention or discovery, is a poor way of exploring the social and cultural (co)evolution of mathematics and methodical experimentation and observation.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#275A 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.
He describes stories as 'relics'. Here's a direct quote from the book: “Stories are relics, part of an undiscovered pre-existing world. The writer’s job is to use the tools in his or her toolbox to get as much of each one out of the ground intact as possible.”
The book is pretty good, well worth the read if you're only slightly interested in the topic!
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#276Earlier quoted context omitted.
And who/what created God -- e.g. who is God's God? And so on? It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.
Why is there a need to insult anyone when it is you yourself that lacks understanding of the matter. We believers believe that God was uncreated, He is unlike His creation. Only created things have a creator. And God created time itself, so there is no before or after Him. He was always there. So there's no other answer than "that's impossible" to your question. Of course this is hard to understand, but what would yo…
> If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there?
The evidence says that actually did happen. Abiogenesis and the evolution of human beings were just part of the process of producing an iPhone through a "mere accident" of natural forces over the course of the last 4.5 billion years or so. Producing an iPhone was never the goal, of course–natural processes don't have goals. But it was one of the consequences.
If God can exist without being created, so can what you refer to as "His creation". This idea that the visible universe requires a Creator, who in turn does not require a Creator, is arbitrary and capricious.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#277A bit both: e.g. The circle is an invention. But the number pi (it's value) is discovered
What do you mean when you say that the circle is an invention rather than a discovery? Are you drawing a distinction between the mathematical ideal and the arbitrarily close approximations which can be found in nature? Though by that criteria that case pi must also be an invention…
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#278Earlier quoted context omitted.
Not at all, pure math is in part about exploring axiomatic systems that may or may not have a physical counterpart. The latter is immaterial.
Can you give some examples of axioms in pure math that run completely counter to our physical world? For example: It is NOT possible to draw a straight line from any point to any other point. It is NOT possible to extend a line segment continuously in both directions. etc... or Things which are equal to the same thing are NOT equal to one another. If equals are added to equals, the wholes are NOT equal. The whole is…
Most of large cardinal axioms.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#279Earlier 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…
"two of our own species [discovered calculus]"..? Which two species are those? I assume by "own own" you mean Terran species, and one of them is Homo Sapiens, which is the other one? Really curious...
Probably Homo Econimus. If Homo Econimus is good at one thing, it's math, to the exclusion of pretty much everything else.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#280Earlier 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…
Maybe vision is helpful with elementary geometry. I don't know how vision would help with differential geometry, geometry of affine spaces, topology, algebraic geometry. Maybe visualization is more helpful, but you can visualize things without seeing. A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required…
Category theory is thoroughly visual, though.