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

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

#221

Post by cwzwarich » Fri, Apr 17, 2020, 4:42 PM UTC

mikorym wrote: post #16

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…

The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic. But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.

> But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around.

There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.

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

#222

Post by mikorym » Fri, Apr 17, 2020, 4:47 PM UTC

cwzwarich wrote: post #221

mikorym wrote: post #16

Earlier quoted context omitted.

The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic. But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.

> But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.

You shouldn't be mapping propositions, you should be mapping theories. See my comment below on the internal logic of a topos.

What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.

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

#223

Post by dorgo » Fri, Apr 17, 2020, 4:48 PM UTC

lisper wrote: post #216

dorgo wrote: post #210

Earlier quoted context omitted.

>The apparent distinction arises from our separation of physical and mental processes. You make the distinction based on (physical or mental) processes involved in the discovery/invention. I wouldn't care so much about how the discovery/invention came to be and look more into what the discovery/invention is about. Usually, you can discover things which are present in the real world. Mathematicians are rarely interest…

But mathematicians are present in the real world, so you can choose to see math as the discovery of the behavior of the brains of mathematicians.

lol, yes, you discover how mathematicians think, but what they think is still an invention.

In the end, I agree that if we expand the definition of the universe beyond physical world to contain "everything" then yes, every invention is also a discovery.

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

#224

Post by Wistar » Fri, Apr 17, 2020, 4:50 PM UTC

Dr. Hannah Fry's 3-part BBC documentary series, "Magic Numbers: Hannah Fry's Mysterious World of Maths," explores this very question in the first episode, "Numbers As God."

The episode description reads: "Documentary series in which Dr Hannah Fry explores the mystery of maths. Is it invented like a language, or is it discovered and part of the fabric of the universe?"

https://www.bbc.co.uk/programmes/b0bn6wtp

She interviews a number of prominent mathematicians and scientists, such as Brian Greene, and they certainly don't agree one way or the other in the invented/discovered question.

(Alas, I now see that the series is listed as unavailable on the BBC site but I watched it, I think on Amazon Prime or maybe youtube.)

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

#225

Post by polytely » Fri, Apr 17, 2020, 4:53 PM UTC

deltron3030 wrote: post #183

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…

>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 influencing the conditions which brought us to life Free will doesn't depend on the universe being deterministic or not. Things that haven't happened yet are likely going to happen in a certain way based on the trajectory, but it doesn't mean that you c…

>but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.

This is of course debatable, there is some evidence to suggest that the feeling that you took a certain path is illusionary. Your brain made the decision, then you became cognizant of the options and you felt yourself making the decision[0], but if you could rewind the universe, you would always choose the same path based on your 'brain-state' at that time. At least that is what I understand of the most extreme "free-will doesn't exist" position.

Just got lost in the SEP rabbithole [1], still not sure where I land on this issue.

[0] : like how you feel thoughts 'bubble up' during meditation, you didn't actively 'think' those, they appeared to you. EDIT: atleast that is what it feels like

[1]: https://plato.stanford.edu/entries/freewill/#DoWeHaveFreeWil...

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

#226

Post by cwzwarich » Fri, Apr 17, 2020, 4:53 PM UTC

mikorym wrote: post #222

cwzwarich wrote: post #221

Earlier quoted context omitted.

> But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.

You shouldn't be mapping propositions, you should be mapping theories. See my comment below on the internal logic of a topos. What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.

Why do you think the same thing doesn't apply to theories? Since you mention topos theory, the double-negation topology is actually a basic construction in topos theory, used to construct a dense Boolean subtopos of a given topos.

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

#227

Post by naasking » Fri, Apr 17, 2020, 4:56 PM UTC

throwaway17_17 wrote: post #195

naasking wrote: post #171

Earlier quoted context omitted.

> Free will is a philosophical concept which is essentially dualist. It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.

First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement. Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it. Pulling a quick quote, the Compatibilist position can typically exp…

> First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement

Almost 60% Compatibilist, the remainder evenly split among three other options: https://philpapers.org/surveys/results.pl

> Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.

You're assuming this is relevant. Turns out, it's not.

> Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it.

Nobody definitively understands free will. Some people conjecture it has certain properties, mainly incompatibilists. So far they have mostly been wrong.

> This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.

Experimental philosohy suggests pretty definitively that people's moral reasoning agrees with Compatibilism: https://www.researchgate.net/publication/274892120_Why_Compa...

> that even if it was supported, that said appeal to authority is a valid refutation of an argument.

The other poster provided no argument, they just made an unsupported claim.

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

#228

Post by naasking » Fri, Apr 17, 2020, 4:57 PM UTC

maxiepoo wrote: post #192

naasking wrote: post #168

Earlier quoted context omitted.

> 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 pr…

I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.

I was merely pointing out that the "mental" dependency of intuitionism isn't strictly necessary. Take it out, and you're left with an equally valid constructive logic.

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

#229

Post by Balgair » Fri, Apr 17, 2020, 4:58 PM UTC

dorgo wrote: post #219

lisper wrote: post #207

Earlier quoted context omitted.

> invention has an element of arbitrariness or choice in it that discovery does not Yes, I can understand how someone could reach this conclusion. I used to believe it myself. But if you think about it you will realize that this appearance of "arbitrariness or choice" is an illusion. Take the airplane for example. Was it invented or discovered? Is there a relevant distinction between the "invention" of the airplane a…

> "arbitrariness or choice" is an illusion after reading a little about quantum mechanics I'm not so sure any more. >Even the arts can be seen as a process of discovery, specifically, the discovery of what processes and artifacts produce favorable reactions in human brains. That assumes that the artists test their work against real audience and adjust their style accordingly. Maybe they do, but I'm sure that there ar…

>> "arbitrariness or choice" is an illusion

>after reading a little about quantum mechanics I'm not so sure any more.

Even Brownian motion would suffice, a chemical phenomena we've known about for millennia.[0] The jittering of sodium and potassium ions between neurons can be enough to trigger spike firing in a 'random' way[1]. At the end of the day, we are just complicated salty sacks of stochastic processes trying to reproduce, not much more.

[0] yes, it's 'quantum mechanical' in cause, like all things are. I'd say it's more statistical mechanics than quantum though, but to each their own: https://en.wikipedia.org/wiki/Brownian_motion

[1] https://en.wikipedia.org/wiki/Principles_of_Neural_Science

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

#230

Post by Phenomenit » Fri, Apr 17, 2020, 4:58 PM UTC

hliyan wrote: post #90

Phenomenit wrote: post #84

Earlier quoted context omitted.

I've always figured it's our visions ability to perceive wholes, one of something. A clear conceptual border between two entities.

Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).

Agreed, I wonder if the same amount of abstract concept would be possible though, can you feel a graph? could you do trigonometry with just hearing and sense of touch?