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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]

#171

Post by naasking »

TheOtherHobbes wrote: post #126
peterburkimsher wrote: post #94

Earlier 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…

Free will is a philosophical concept which is essentially dualist. It's an attempt to reconcile two subjective experiences - the feeling of making a decision, and the fact that sometimes we can't predict the actions of others - with universal determinism. There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will wi…

> 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.

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

#172

Post by symplee »

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.

Isn't it the other way around? Axioms are chosen because there are no observable counter examples in the real world.

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

#173

Post by analbumcover »

ironSkillet wrote: post #146
analbumcover wrote: post #129

Earlier quoted context omitted.

It's the other way around, classical logic is a restriction of intuitionistic logic that arises when you affirm the Law of Excluded Middle.

I think you're both right in a certain sense. The set of provable statements in intuitionist logic is a subset of what is provable in classical logic, which is what I believe the parent was referring to.

You're certainly right that any constructively provable statement is classically provable, that follows from intuitionism/constructivism generalizing classical logic. But I don't see how we could interpret the following to mean that.

> you can do intuitionist mathematics in classical mathematics

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

#174

Post by naasking »

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.

> If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered

How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis.

> The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.

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

#176

Post by Reedx »

Same question also answered by

Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8

Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4

Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM

I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

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

#177

Post by ryugarden »

cpsempek wrote: post #56
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…

I think I get your point and I‘m open to ideas which drive home your point in a more precise way. However, the main difference between concerning ourselves with whether or not mathematical objects exists vs music or start-up ideas is their place in building a foundation for understanding reality. Science has been a productive program for understanding reality. It happens to be underpinned by math. And as a result, we…

>do mathematical objects exist?

Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic.

>the main difference between concerning ourselves with whether or not mathematical objects exists vs music

There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way. For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead).

So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example: https://youtu.be/QzBoGVToWEo?t=342

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

#178

Post by naasking »

Tainnor wrote: post #156
otabdeveloper4 wrote: post #143

Earlier quoted context omitted.

> 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…

> 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.

Except it's not ineffective at all in these subjects. Social science experiments have so many variables that the small experiments that can be conducted given the financial resources are insufficient to infer a good model. This is not a math problem, it's a money problem.

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

#179

Post by 7thaccount »

Reedx wrote: post #176

Same question also answered by Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8 Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4 Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

Max Tegmark has some fun books. Apparently he got to be at the forefront of some discoveries as he and his wife knew C++ and would do all the after the fact data analysis.

If you've ever been around a campfire with a good storyteller that has their audience at the edge of their seat, I felt like this in some spots.

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

#180

Post by pron »

analbumcover wrote: post #142
pron wrote: post #42

Earlier quoted context omitted.

Patents cover inventions that are applications of knowledge. For example, chemistry is certainly discovered, but applying the use of a certain molecule as a glue or as a drug is very much patentable. Physics is also discovered, but a special lever system that is a direct application of Newtonian mechanics is patentable. The knowledge that a certain mathematical formula could be written and has certain properties is n…

I don't see how you are distinguishing between inventions and discoveries. The "discovery" of a mathematical proof is just an application of knowledge, why aren't they patentable? The "invention" of the light bulb, on the other hand, is just the discovery that putting various particles together in a certain configuration produces light in a predictable manner, why is this patentable? I don't see how a clear delineati…

I'm not. I'm saying that devices whose function is derived from mathematics (algorithms) have the same status as devices that derive their function from physics or chemistry. In either case, "application" doesn't mean some intellectual use but commercial use. The purpose of patents is to protect commercial applications in exchange for sharing the knowledge that led to their creation. The knowledge itself is never protected.

When the lightbulb was patented, everyone was free to learn, study, and disseminate the physical knowledge of how a lightbulb works. What you couldn't do is build one and sell it. Whether mathematics is invented or discovered, the knowledge itself is never protected, but if some non-obvious algorithm has a commercial application, it can be patented so to protect building devices that apply that knowledge for commercial use.

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