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The Paradox of the Proof

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Re: The Paradox of the Proof

#111

Post by qznc »

gwern wrote: post #101
ics wrote: post #84

Earlier quoted context omitted.

> Is there any way of even roughly estimating how much Flattr might bring in? Of course: you try it, write up a massive future post about the results, and voila! But that probably wasn't the answer you wanted. I have no idea how many people are using it, but the idea sounds pretty great and if there's a good crowd to try it out on then HN readers wouldn't be a bad bet. I don't think anyone would mind another small bu…

> Of course: you try it, write up a massive future post about the results, and voila! ~-~ > I don't think anyone would mind another small button between PayPal and BTC. Yes, fair enough... I think I may just be making excuses at this point to not try it out.

Just put it in my website footer [0]. Flattr is very flexible. You can use various methods with more or less or no JavaScript. Also you can link stuff like Github, so starring a repo leads to flattr compensation.

[0] http://beza1e1.tuxen.de/articles.html

Re: The Paradox of the Proof

#112

Post by maxerickson »

dredmorbius wrote: post #99
calinet6 wrote: post #33

Earlier quoted context omitted.

I wonder what would happen if, say, I allocated $10 per month to compensating articles and journalists I enjoyed. And then it was distributed evenly by time I spent on the page, every single day. So if I spent 30 minutes reading this article, 10 minutes reading another, 2 minutes on 10 others, this author would get $0.16 ($0.33 per day, times (30 mins, out of 60 total mins of time = 0.5)). That's really not that much…

Readability tried something like this. Among the problems: authors / publishers didn't sign up, and money collected had to be distributed or reimbursed. Unclaimed funds eventually were awarded to a charity. Though I believe Readability had good intent, they took a great deal of flack for this. Money weirds things.

Most of the weirdness in the case of Readability came from charging their users before they had established the relationships to pass the funds along. Collecting funds for authors sort of advertises that you are paying them, doing it while not being able to pay them is pretty shady.

I guess a better way to bootstrap such a thing would be as part of some sort of time tracker, show the user where they spent their time at the end of the month and wire it up to a tip jar system.

Re: The Paradox of the Proof

#113

Post by ky3 »

neogodless wrote: post #74

As a completely uneducated simpleton, it seems bizarre to me that addition and multiplication are considered "different" in the deeper explorations of math and number theory. It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be repr…

It seems like multiplication is just an extension of addition. How many times do you want to add numbers together? The result is multiplication. Similarly, addition can be used to represent multiplication. You want to multiply, which can be represented as adding things a certain number of times.

If you're interested in this train of thought here's a resource you might want to chase up:

http://en.wikipedia.org/wiki/Hyperoperation

As for the "bizarre" interplay between addition and multiplication, you might want to contemplate The Other FLT [1]. TOFLT can be understood with just high-school math but is a big gateway drug into big-ass number theory. In more ways than one!

[1] http://en.wikipedia.org/wiki/Fermats_little_theorem

Re: The Paradox of the Proof

#114

Post by rickhanlonii »

ricardobeat wrote: post #104
brazzy wrote: post #27

Earlier quoted context omitted.

> It is reality what drives mathematicians to work, not a pure abstraction. In general, no. > It is because of reality that maths is interesting. It is because of Geo-metry that algebra is so relevant. Perhaps to you, but not to most mathematicians, certainly not those working in academical settings. Have you ever looked at group theory or topology? I don't think you've ever done the kind of math that mathematicians…

> invent constructs that have no basis in reality Regardless of how abstract you get, how far you go, math is always tied to our physical reality. The basic operations are reflections on properties of our universe. The "kind of stuff" mathematicians do" allows us to model and reason about our world in ways that wouldn't be possible any other way (that we know of).

>math is always tied to our physical reality

I'm sorry but even if this is objectively valid, this not a claim that you can support. There are entire sub-fields of maths in which there are no known physical attachments.

Now, that's not to say that some time in the future we won't discover the relationship between every mathematical concept and some physical system. But at this moment, the claim you are making has a numerable set of counter examples, with a very large order. For one example, take the Banach–Tarski paradox:

>Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.[1]

This is, as far as we know, physically nonsense.

That being said, there is quite a lively debate among mathematicians as to whether or not math is tied to reality.

[1] http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Re: The Paradox of the Proof

#115

Post by FoeNyx »

Rexxar wrote: post #53
brazzy wrote: post #43

Earlier quoted context omitted.

> the only people who exhibit this type of behaviour are frauds and delusional pseudoscientists. But those usually produce work that is very obviously nonsensical, and want attention rather than avoiding it. Mochizuki has earned the privilege of having his work evaluated on its own merits, and so far nobody has found any obvious flaws. > And the only rational reason to do so is if there is actually something wrong ab…

> Can you give some examples? Isaac Newton : http://en.wikipedia.org/wiki/Isaac_Newton%27s_occult_studies

Except he's from the pre-modern era and does not fit the definition of what we call now a "scientist". He probably did science (laws of motion, universal gravitation, etc) alongside alchemy, and occult studies.

It's rather a counter example : you can do occult studies, while in the same time providing the world some of the most revolutionary scientific concepts.

Re: The Paradox of the Proof

#116

Post by rickhanlonii »

andreasvc wrote: post #93
raverbashing wrote: post #20

Earlier quoted context omitted.

" you need to abstract away from standard preschool algebra and build up models that then let you get said general results" Yes, of course. " but could you maybe still make a suggestion on how to better think about problems?" And that's what I meant. Thinking about problems in a different way (but still provable, and still working in a similar way) For example, for Peano arithmetic you have that equality is symmetric…

Your suggestion reminds me of typographical number theory (TNT), in Hofstadter's Goedel Escher Bach. TNT is introduced for didactic purposes, I don't see the point of using it for actual mathematics.

That's what I thought when I read his comment. Hofstadter emphasizes how unmanageable such a system becomes quite quickly.

For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij

Re: The Paradox of the Proof

#117

Post by rickhanlonii »

pfortuny wrote: post #42
30thElement wrote: post #22

Earlier quoted context omitted.

As someone with a degree in applied math, the pure abstract is more interesting than the applied. Applied math is like building really amazing and intricate sand castles on the beach. Pure math is like building the same sand castle, but in the sky and it's kept aloft purely by how beautiful it is, freed from constraints like "touches the ground" and "can support itself under gravity". A lot of my friends feel the sam…

Look, I am an Algebraic Geometer and have done Schemes and whatnot. It is REALITY and this has nothing to do with "applied" or "pure". The fact that it is abstact has nothing to do with its being unreal. Just to clarify: I am an expert too.

Pure math evangelist here. To be clear, it is only reality insofar as the axioms in which the theorems are derived are reality, and only insofar as our understanding and application logic is an objectively valid construct of reality.

When it all works out as beautifully as it does, in say Euler's Identity, it's hard to remember the possibility that the axioms could turn out false, or logic as we know it flawed.

But assuming (heh) that the axioms are true and that our understanding of logic is valid, "pure" math is as much apart of reality as "applied" math. And I'll choose proving the Fundamental theorem of Galois theory over number crunching in Matlab as my exercise in experiencing reality every time.

Re: The Paradox of the Proof

#118

Post by jules »

voyou wrote: post #51
spitx wrote: post #26

"For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it." I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension. The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or…

It's cool that "Alex Knapp, a science writer at Forbes" has solved one of the major issues in the philosophy of mathematics by, erm, just asserting that Platonism is false. Of course, maths is a tool, but that doesn't mean that it is just a tool, and Knapp doesn't provide any arguments for that conclusion. I can use a rock as a tool, but that doesn't mean the rock has no existence independent of my mind. In fact, the…

I find what's written on that page much more naive than Knapp's view. Knapp is not advocating fictionalism, i.e. that math has no meaning. He is advocating the opposite: math has meaning, but the reason is not some metaphysical mystery, it has meaning because it was constructed for that purpose. Knapp's view is much closer to what that page describes as physicalism, with the caveat that humans are not perfect so our math is not necessarily the same as alien math.

That is also one of the instances where that page is so naive. It dismisses physicalism in a just a couple of words on the grounds that physicalism cannot explain infinities, especially very large infinities. But mathematicians themselves are divided on whether those are "real" or just an artifact of the axioms (see constructivism and more radically finitism). Another view that fits perfectly within physicalism is that infinities are just an approximation, a reasoning tool to simplify things.

Re: The Paradox of the Proof

#119

Post by raverbashing »

rickhanlonii wrote: post #116
andreasvc wrote: post #93

Earlier quoted context omitted.

Your suggestion reminds me of typographical number theory (TNT), in Hofstadter's Goedel Escher Bach. TNT is introduced for didactic purposes, I don't see the point of using it for actual mathematics.

That's what I thought when I read his comment. Hofstadter emphasizes how unmanageable such a system becomes quite quickly. For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij

Yes, I was thinking of it when I wrote the comment, there's also an arithmetic with P an M symbols - for plus and minus (but I can't find it on google, it's been a while, sorry)

Thanks for sharing this snipped, one of the several fun things in GEB

Re: The Paradox of the Proof

#120

Post by rickhanlonii »

raverbashing wrote: post #119
rickhanlonii wrote: post #116

Earlier quoted context omitted.

That's what I thought when I read his comment. Hofstadter emphasizes how unmanageable such a system becomes quite quickly. For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij

Yes, I was thinking of it when I wrote the comment, there's also an arithmetic with P an M symbols - for plus and minus (but I can't find it on google, it's been a while, sorry) Thanks for sharing this snipped, one of the several fun things in GEB

No problem, the image at the beginning of chapter 10 is one of my favorite GEB gems: http://i.imgur.com/NahfzTp.jpg
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