Roger Penrose – Is Mathematics Invented or Discovered? [video]
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Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#2Intuitionist 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 to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1]
[0] https://en.wikipedia.org/wiki/Intuitionism
[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#3The SEP article is an excellent introduction to the philosophy of mathematics:
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#4Intuitionist 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…
> Applying intuitionist mathematics to physics we can come to the conclusion that time flows
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#5Intuitionist 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…
From the wikipedia source:
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#6Recently I watched Sixty Symbols video on why light is slower in glass and there are multiple explanations on it.
What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk
On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality.
That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable.
Wouldn't that mean that mathematics is just an invented tool to reason about physical models?
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#7Of course..we invented the ‘language of mathematics’.
I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind Audiobook.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#8Put me into the 'discovered' camp.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#9The expression is invented but the underlying relationships are discovered.
It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter.
It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.
Re: Roger Penrose – Is Mathematics Invented or Discovered? [video]
#10Recently I watched Sixty Symbols video on why light is slower in glass and there are multiple explanations on it. What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is sim…
Well yes and no. A lot of mathematics found its roots in trying to model the real world, however I would then argue that the truths we prove about these models are truly discoveries. And with these discoveries we can often generalize the setting to more abstract formulations, independent of the physical reality.
I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical.
This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.