The Paradox of the Proof
projectwordsworth.com
projectwordsworth.com
There are interesting parallels and contrasts with Thomas Hales’s proof of the Kepler conjecture. In that case, as far as I know Hales did everything possible to help his colleagues understand the proof, but even so it was so long and involved that the referees declared themselves unable to be certain it was correct. Since then, he has been working on a formal machine-verifiable version of the proof under the banner of the Flyspeck Project: http://code.google.com/p/flyspeck/
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. With 10,000 people all following this same path, the author would "only" get $1600. Readership is likely to drop off exponentially after the initial publicity. Publishing two articles of similar quality per month would get the author a nice living wage; not too bad, but only if they can reach an audience of 10,000 interested readers who are in on this compensation system. Slim chance.
Perhaps it works as a supplementary reward system. Another question: does this incentivize the right things? Will such metrics lead to longer articles which aren't necessarily interesting to read, but just take a long time? Does it disincentivize quality in any way?
More questions: how much would people be willing to contribute—$10 per month? $30 per month? How is it distributed—manually or automatically, by time or by rating? Is the small bit better than nothing? Does this incentivize people to a) produce good content, or b) pay for its consumption? Would people voluntarily buy into this system, or does it need to be a restrictive thing where only contributors can read articles?
Just thinking.
It's actually quite successfull in the German web community especially for podcasters. There are German podcasters which earn about 1000 Euro per month via Flattr.
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 button between PayPal and BTC.
~-~
> 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.
Though I believe Readability had good intent, they took a great deal of flack for this.
Money weirds things.
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.
I can see this being the management team getting out in front of itself though. I can understand how others would criticize. I don't see any obvious signs of intentional malfeasance. Perhaps poor judgement.
The solution they proposed is one that I've been thinking should be applied to online content for some time.
Then one guy solved it. The changelist he prepared wasn't even that long but it was crazy complex. Helpful comments contained links to a 100-pages long academic paper on top trees. Since it had been long established on the team that the guy was (a) a genius and (b) hundred percent reliable, it was generally assumed that the solution would work, however all code changes at Google need to be reviewed and nobody was able to review that CL. A number of people tried but they all dropped out, even though, unlike the ABC Conjecture professor, the author was absolutely willing to answer any questions you had.
Eventually the powers that be decided to trash the CL, not because there was any doubt regarding its correctness, but on the principle that you can't let anything that only one person can comprehend enter the codebase.
You don't have to write something that is essentially unreviewable to end up in the scenario described earlier. You just have to work with a team that isn't prepared to learn new things and rejects thing that they don't understand.
I've been studying functional programming lately and can easily imagine being told my code is "incomprehensible" because I wrote it in a functional style instead using loops and variables.
All the warning signs of pseudoscience are there: "I couldn't possibly explain it" as a response to lecture invitations, working on something for extended periods of time without sharing results, using lots of obscure terminology which is not standard for the field, one long and obfuscated paper instead of building toward the result incrementally, etc. Using the title "Inter-universal geometer" instead of calling yourself a mathematician is also strange.
The natural way for a mathematician to behave after seemingly solving an important problem is exactly the opposite of what this guy is doing. And the only rational reason to do so is if there is actually something wrong about the work.
I lack the know-how to arrive at my own opinion by reading the paper, but this situation is definitely fishy. And it's not like a bona-fide scientist losing it and becoming an pseudo-scientist obsessed about a topic is completely unheard of.
So, yes, something is fishy, but so was the weird idea of curved spacetime.
I agree it seems odd but even in my limited exploration into Pure Mathematics I have seen alternative proofs made with ideas not native to the field. Why must that necessarily make this proof invalid?
My only issue is that in the article, and in other writing on the subject, nobody is even contemplating the possibility that Mochizuki's work might be unreadable for reasons other than it being too brilliant to grasp.
I wanted to present an alternative possibility which seems to be disregarded at the moment in favor of the attractive "eccentric genius" narrative.
On the topic of Perelman - he did reject the Fields Medal. However, he did give a series of talks at MIT, Princeton and other places a year after publishing his proof.
So to me, that read more as, this is a curiosity that has a deep and interesting past and an even more interesting present.
> I wanted to present an alternative possibility which seems to be disregarded at the moment
That is entirely fair and valid. I just misread your comment as more along the lines of "Does no one else see how obvious it is that this guy is crazy?!" My fault.
> On the topic of Perelman
Yes I was already corrected. I couldn't remember with certainty whether he did or didn't. I thought he had, but then what I remembered about his personality made me reconsider that.
At the same time, it's important to check the work rather than just dismissing it. Mochizuki's work isn't totally original - a lot of it is derived from existing theory. So it should be possible to validate it despite the complexity.
It's true that serious breakthroughs from individuals working alone are rare, but they are not unheard of. The real test is whether or not this work can be adapted to the rest of mathematics (or the other way around, perhaps).
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 about the work.
Who says the reason has to be rational? Maybe he just has developed a bad case of stage fright?
> And it's not like a bona-fide scientist losing it and becoming an pseudo-scientist obsessed about a topic is completely unheard of.
Can you give some examples?
Isaac Newton : http://en.wikipedia.org/wiki/Isaac_Newton%27s_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.
Mathematics is the same, to an extent; one guy working alone for 14 years is likely to have missed ideas and perspectives that could illuminate flaws in his reasoning. Maths bugs. If he's produced hundreds of pages of complex reasoning, on his own, however smart he is I'd say there's a high chance he's missed something.
Humans need to collaborate in areas of high complexity. With a single brain, there's too high a chance of bias hiding the problems.
(Repost of my previous comment https://news.ycombinator.com/item?id=4829806)
edit: more constructively, Imagine that you're working on the NYT crossword and I come along and point out that 45 across is wrong and tell you what the answer should be. Do you then throw away the rest of your work? No, you fix the part that's wrong and then check the rest of the puzzle to figure out the scope of the error.
It did have a flaw that got fixed later in collaboration, but most of the work was one individual's deep focus.
"...Mochizuki is holding a private seminar with Yamashita, and Kim hopes that Yamashita will then go on to share and explain the work."
The issues are complex. Mochizuki apparently has some diffidence about communicating with the wider mathematical community. Yamashita may have to act as spokesman (and an initial checker). Then once communicated the checking and sifting can begin (and the recycling of new tools start).
Assuming the work is intelligible and valid of course.
Um, no. Mathematics itself has absolutely nothing to do with "describing the universe". Mathematics is purely abstract. It has no inherent relationship whatsoever with reality.
That certain mathematical constructs can be used to model certain aspects of the real world is basically a lucky coincidence, but not very interesting to mathematicians - that's what physicists do.
David Deutsch has a method for ascertaining whether something can can be said to exist or not - ask whether it "kicks back" when you interact with it, in the sense that simulating the response of the thing you're considering in a totally convincing way would involve an effort as large as building a new universe for that thing to exist in.
Rocks "exist" because, if you wanted to build a totally convincing simulation of a rock, you'd need to include all of modern physics as we currently understand it.
Other minds "exist" because simulating them convincingly would basically require you to construct a true articificial intelligence (strong AI).
Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity.
There is a very real truth in the statement that '1 + 1 = 2', or the statement that there are arbitrarily long arithmetic progressions of prime numbers, or any number of other mathematical results. The world of mathematics is a very real part of the universe, that "kicks back" by constantly surprising us when we think about it.
So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.
Well, one could say that we "create" them using our minds, which certainly are, but that's deep into philosophical territory.
> Mathematics "exists" because giving someone the genuine experience of doing mathematics when they really weren't would involve a simulation of almost unfathomable complexity.
Sounds to me like a misapplication of the method - the complexity arises from simulating the response of a person to math, not from simulating math. By that standard, all fiction is "a very real part of the universe".
> So I don't think the article, which is very well written and researched, deserves the middlebrow "Um, no" scorn that you treated it to.
That concerned on one statement, not the entire article (which I found fascinating as well). Yes, I have to admit that this is rather smartassy, but I actially feel that, quite independant of this article, it is an important and amazing realization that few people make.
Strangely, I believe that statement supports crntaylor's argument, being the point of fiction as far as I can see.
oh, sure, but everything is part of the universe. human thought, for instance, is part of the universe. and i'd say that's what mathematics is about.
What if there are 0 universes? Does 0 exist without any universes?
Not to speak of Calculus...
To many mathematicians reality is largely irrelevant. For example G. H. Hardy, an extremely prominent British Mathematician, believed that true Mathematics is an art form and is not useful. He dismissed applications of mathematics as dull and boring.
And I can relate to him. It's incredible how complex, beautiful structures arise from a couple of simple axioms. It doesn't matter if what you study will be relevant or not, what matters is that it's fun and stimulating to explore.
A lot of my friends feel the same way, with some of them specifically avoiding having "real world" applications of their work, as if that makes it an even better sand castle.
As to why I have an applied degree instead of doing pure math, numerical analysis makes a weird intuitive sense to me, and I figured building decent sand castles on the beach was better than making terrible sand castles in the sky that could barely hold themselves up. It also gets the grant money.
Just to clarify: I am an expert too.
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.
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 do. What you learn in school is not math, it's calculating. What mathematicians do is to invent constructs that have no basis in reality and prove statements about them, then come up with more constructs based on those statements, ad infinitum.
Sometimes those constructs may be designed to model real world problems, and getting funding is probably easier in those areas, but just as often the applicability is only discovered afterwards - or not at all.
The best example (because it's something we've actually all learned about) is complex numbers. They were first invented in teh 16th century and considered pointless and irrelevant at first. People soon discovered that they could be useful in proofs about non-complex numbers as well, but it took several centuries before they were found to be directly applicable in electrical engineering (many more applications have been discovered since).
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).
No, this is most definitely not true for all branches of mathematics.
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
I agree that's not the literal sentence you quoted, but it's definitely the theme of the article: according to many of the people cited, Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect.
Caveat: I Am Not A Mathematician (IANAM)
Understanding is necessary to be certain the proof is correct. After that is ensured, many mathematicians will happily use the proof and base their work on it without understanding it.
> Mochizuki's shirking his responsibility by not explaining how to understand the result, regardless of whether the proof is correct or incorrect.
The main problem caused by his "shirking" is that it leaves the correctness of the proof uncertain and trying to fully understand it is a lot of work that could turn out to be wasted. People want to confirm the proof, and Mochizuki could make it much easier and less costly for them, but for some reason declines to do so.
edit: I can see taking issue with the word, "universe" in the original quote. I have trouble seeing how the word "understand" is debatable.
This book has an extremely modern discussion of this
http://www.phil.cam.ac.uk/teaching_staff/potter/staip.html
Only lay people and arm chair philosophers passing through math land have this unfortunate view.
Also, i find it quite surprising that the proof for problems in domains as elementary as number theory, should have to be so complex, sort of baffles me. I hope i can rise up to the level to begin to understand this lingo or that someone brings it down to the level where i can find it interesting to read, like this article :D
One reason for this is that primes are defined by their multiplicative properties. There aren't many easy connections between primes and the additive properties of numbers, and so proofs that try to relate additive things to primes tend to involve deep and complicated things.
And here is a FAQ on the theory: http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-U...
http://today.uconn.edu/blog/2012/10/the-mochizuki-theorem-wh...
http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-be...
That's why a lot of mathematicians are sceptical against computationally constructed proofs such as state space exploration. Or, at least, they don't like the taste of it :)
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.
Of course, the conjecture introduces rules about prime numbers, and then says "ooo" now we see that prime numbers being added together (with rules of what kinds of prime numbers are allowed in the equation) results in "predictable" rates of incidence.
I guess it's a little late to go back and be born again with a life of education centered around number theory so I could see and comprehend the complexity!
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.
It might be easier to think of this with regard powers. People teach that A^5 is just AxAxAxAxA. You then deduce that A^a x A^b = A^(a+b). From that you start to assign meanings to things like A^0. And A^(-1). But what does it mean to multiply together -1 copies of a number? That doesn't make sense!
And what about A^{\pi} ? How can you have a transcendental number of things multiplied together? It doesn't make sense!
As you get deeper into math you need different definitions of powers, and of multiplication, and you find they they coincide with repeated multiplication and repeated addition, they may have originated with those ideas, but that's not really the best way to think about them, and it's not, in some sense, what they "are".
A poor analogy might be this. To an outsider, Smalltalk and Haskell will kind of look the same. They're programming languages, they do the same things. But they are really very different animals. So multiplication is really a very different animal from repeated addition.
You see the same pattern in programming. Imagine if you had to have separate addition operators for doubles, floats, ints, and bigints. It's far more useful to have one function that only depends on the behavior of its input, rather than having its input be a very specific type of object.
> Similarly, addition can be used to represent multiplication.
Yet, addition cannot be represented as multiplication (AFAICT), so there you have a difference.
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] described in this paper: http://www.jstor.org/discover/10.2307/4182081?uid=3739856...
I wish I could find the animation itself, but that link is broken.
If you find a copy of his explanation in Japanese I'll try to see if he's referring to a literal animation or not. I couldn't find one and he doesn't really refer to anything beyond the theme/concept and the girl's theta-like eyes.
Edit: Something else I found (https://www.jstage.jst.go.jp/article/essfr/6/3/6_160/_pdf).
I didn't really get much from it even with some assistance, but the very first part is a quick dialogue about a universe king and a neighbor universe king exchanging New Years gifts. Section 7 is about the そっくりハウス and goes with the page you linked, but ends by saying something about the multiplicative and additive rotational properties have something to do with the different universes seeing each other. He compares the girl's excitement to Mochizuki's upon discovering the identical house inside of a house.
Someone with better command could give a much better summary, but I'd personally be more interested in the lecture Mochizuki gave himself on it IUTeich last week (according to his site).
I think mathematicians have a weird way of thinking about problems.
First-order logic for example: http://en.wikipedia.org/wiki/First-order_logic
It's quirky to think, for example, on the natural numbers that 'exists an operation + and a null element under that operation 0'
This is "very understandable" by humans, but very difficult to compute.
As such as this conjecture is stated in a way that looks qualitative (but has a good definition), still, usually it seems that proofs are even harder for theorems defined like that
You called it a rant, but could you maybe still make a suggestion on how to better think about problems?
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 (and transitive)
Now, there are several ways to explain that, and it's usually explained more or less by "for all X and all Y, if X = Y then Y = X"
Now, it would maybe be interesting to have a 'different explanation' that is as powerful as first order logic but works differently (and maybe easier to compute)
For example, it may be possible to write Peano arithmetic as a grammar (so zero would be ' ', one would be I, two would be II, etc)
Furthermore, I have to admit I don’t see the immediate advantage such a reconstruction would bring with it.
Well, there may not be immediate advantages, but in math you never know. There are several hard problems in one domain that are trivial in another domain, for example.
An example from physics: http://en.wikipedia.org/wiki/Hamiltonian_mechanics
Certainly, and this is pretty much what OP did, invent a new domain to solve a problem – Hamilton aka Lord Kelvin merely reformulated the problem slightly, and while I personally love Hamiltonian mechanics, I don’t think it is comparable to ‘inter-universal geometry’ or replacing first order logic with something else.
So, yes, a different field may provide a different perspective and hence easier solution, but if you want to replace first order logic, you’re not looking at a different/new field in maths, you’re looking at rebuilding maths.
For example, here's what a proof that addition is commutative looks like in TNT: http://imgur.com/a/O63Ij
Thanks for sharing this snipped, one of the several fun things in GEB
I seem to recall doing this as a homework assignment in computer science, writing some unrestricted grammars that could be used to perform certain simple operations on some simply-encoded numbers.
See also http://en.wikipedia.org/wiki/Thue_%28programming_language%29
But I wonder if you've gotten very far into mathematics. It's actually well known that there are many things equivalent to first-order grammar and it's completely common to choose different things based on the sort of thing you're doing, just as many things are equivalent to Turing Machines and one can freely choose the most convenient one for your local problem. Peano arithmetic is chosen merely for its convenience for the simplest of proofs, and the instant it becomes inconvenient a real mathematician abandons it for something more locally useful.
Source: 4th year mathematics student.
School failed at conveying this to me in so many domains.
People tell you that the point of being good at science
and math is to help you understand the universe, and to
understand what's going on.
This is only the first step.
The real reason for being good at math and science is to
BEND THE WORLD TO YOUR WILL !!
"For centuries, mathematicians have strived towards a
single goal: to understand how the universe works, and
describe it."
The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or lack thereof) of mathematics to the fabric of our reality.
Alex Knapp, a science writer at Forbes :
In the midst of a rather interesting discussion of the
notion of Aristotle’s Unmoved Mover, Leah Libresco went on
a mild digression about the philosophy of mathematics that
I couldn’t let go of, and feel compelled to respond to.
She says:
I take what is apparently a very Platonist position on
math. I don’t treat it as the relationships that humans
make between concepts we abstract from day to day life.
I don’t think I get the concept of ‘two-ness’ from
seeing two apples, and then two people, and then two
houses and abstracting away from the objects to see what
they have in common.
I think of mathematical truths existing prior to human
cognition and abstraction. The relationship goes the
other way. The apples and the people and the houses are
all similar insofar as they share in the form of two-
ness, which exists independently of material things to
exist in pairs or human minds to think about them.
The notion that there’s something special about math –
that it has some sort of metaphysical significance – only
makes sense if you ignore the history of how we uncovered
math to begin with. It was, despite Leah’s protestations,
exactly just the abstraction of pairs and triplets and
quartets, etc. The earliest known mathematics appear to be
attempts to quantify time and make calendars, with other
early efforts directed towards accounting, astronomy, and
engineering.
Mathematics is nothing more and nothing less a tool that’s
useful for humans in solving particular problems. Math can
be used to describe reality or construct useful fictions.
For example, we know now that the spacetime we live in is
non-Euclidian. But that doesn’t make Euclidian geometry
useless for everyday life. Quite the contrary – it’s used
every day. You can use mathematics to build models of
reality that may not actually have any bearing on what’s
real. For example, the complicated math used to describe
how the planets moved in the Ptolemaic model of the solar
system – where everything orbited in circles around the
Earth – actually produced very accurate predictions. But
it was also wrong. There aren’t actually trillions of
physical dollars circulating in the economy – there are
just symbols for them floating around.
The bottom line is that human beings have brains capable
of counting to high numbers and manipulating them, so we
use mathematics as a useful tool to describe the world
around us. But numbers and math themselves are no more
real than the color blue – ‘blue’ is just what we tag a
certain wavelength of light because of the way we perceive
that wavelength. An alien intelligence that is blind has
no use for the color blue. It might learn about light and
the wavelengths of light and translate those concepts
completely differently than we do.
In the same way, since the only truly good mathematicians
among the animals are ourselves, we assume that if we
encounter other systems of intelligence that they’ll have
the same concepts of math was we do. But there’s no
evidence to base that assumption on. For all we know,
there are much easier ways to describe physics than
through complicated systems of equations, but our minds
may not be capable of symbolically interpreting the world
in a way that allows us to use those tools, any more than
we’re capable of a tool that requires the use of a
prehensile tail.
Math is a useful descriptor of both real and fictional
concepts. It’s very fun to play around with and its
essential for understanding a lot of subjects. But it’s
just a tool. Not a set of mystical entities.
In short, his explanation allows for some (or a very large number of) mathematical truths (according to the consensus of mathematicians and what appeals to their logic) to be just that -- figments of numerical imagination that neatly sit in the confines of our logic.
Nothing necessitates all mathematical truths (much less conjectures) to be corresponding to some aspect (however minute or however large) of our reality.
Some truths might (purely out of happenstance), but nothing mandates that all math truths correspond to some facet of our physical reality.
So, some (or a lot of) math is just hocus pocus.
Non-mathematicians will never know owing to the very nature of peer-review and the consensus-building aspect of modern research and scholarship.
A few questions:
What other conclusions can be drawn if one were to find this explanation appealing?
Are there other explanations of the relationship between math and our reality, that you've found appealing?
Is there a consensus among mathematicians as to what higher-order math, essentially is in pursuit of or should be in pursuit of?
Is it an exercise of random shooting of darts hoping that some "mathematical truth" sticks and corresponds to some observable phenomenon?
Source:
Does Math Really Exist?
http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-r...
Edit: Clean-up and rewrite.
That even if we find a Grand Unified Theory of Everything that fully explains the physical world, we'll never run out of things to discover.
> Are there other explanations of the relationship between math and our reality, that you've found appealing?
None that makes sense, given what I know.
> Is there a consensus among mathematicians as to what higher-order math, essentially is in pursuit of or should be in pursuit of?
The potential for more math. Mathematicians look for theorems and constructs that are "interesting", which basically means that they are neither simple nor random. Prime numbers would be uninteresting if they were regular, but also if they were random. In fact they are neither, there's a multitude of patterns - the Wikipedia category "Classes of prime numbers" has 69 pages in it!
> Is it an exercise of random shooting of darts hoping that some "mathematical truth" sticks and corresponds to some observable phenomenon?
Yes, but some people don't care where the darts go, while others care about observable phenomena and notice any darts lying near them and start asking around who shot those and what they were doing...
That's odd. In my experience, these detours in these discussions come as a result of how vague this whole terrain is. There is not even a smidgen of agreement on even the basic of agreed things.
Can't mathematicians and theorists not agree on what areas of math are most helpful and what areas least helpful in arriving at a Unified Theory?
This leads up to the other question of consensus among mathematicians.
>The potential for more math. Mathematicians look for theorems and constructs that are "interesting", which basically means that they are neither simple nor random.
Isn't there a faction of math people who strive towards a defined, non-abstract direction as opposed to fostering a laissez-faire approach to mathematics scholarship that naturally creates more math so that their area of expertise gets recognition and not to mention substantial purses of money?
Is math so poorly funded that all mathematicians lead a hand to mouth existence and therefore collectively as some sort of cabal, have to resort to these self-preservation tactics?
Come on. Really?
Not fully, and with good reason (see below).
But actually, my point was that the existence of mathematical constructs that do not correspond to any physical reality means that there will be math to do when (and if) all physics has been done.
> Isn't there a faction of math people who strive towards a defined, non-abstract direction as opposed to fostering a laissez-faire approach to mathematics scholarship that naturally creates more math so that their area of expertise gets recognition and not to mention substantial purses of money?
Yes and no. There is the branch of applied math, and I'm sure they get funding more easily.
But there are also mathematicians (cited several times in the comments here) who see math as art and want to do it for its own sake.
And it has happened quite often that these "pure" mathematicians came up with enirely new stuff that only afterwards (and without anyone foreseeing it) turned out to be useful in modelling physical processes. Even among mathematicians you sometimes find that you can prove something in one field by using constructs and theorems from an entirely different field that nobody thought was in any way related. I believe Wiles' proof of Fermat's Last Theorem was like that.
Here's a review of the "Study Edition" of this book:
"How much mathematics can there be? they are asked.
Instructors in a Mathematical Experience course must be
prepared to respond to questions from students concerning
the fundamental nature of the whole mathematical
enterprise. Stimulated by their reading of the text,
students will ask about the underlying logical and
philosophical issues, the role of mathematical methods
and their origins, the substance stance of contemporary
mathematical advances, the meaning of rigor and proof in
mathematics, the role of computational mathematics, and
issues of teaching and learning. How real is the conflict
between “pure” mathematics, as represented by G. H.
Hardy’s statements, and “applied” mathematics? they may
ask. Are there other kinds of mathematics, neither pure
nor applied? This edition of the book provides a source
of problems, collateral readings, references, essay and
project assignments, and discussion guides for the
course."
"The authors state, “Most writers on the subject seem to
agree that the typical working mathematician is a Platonist
on weekdays and a formalist on Sundays.” The substance
of the mathematics appears to change with experience and
depends on the person recounting the story. But it has an
objective reality that is independent of the person. Alas,
when precision is required, it is common to retreat to the
formalist position that mathematics is only a created
structure of axioms, definitions, and their consequences."
http://www.ams.org/notices/199710/comm-millett.pdf ( PDF )
http://www.springer.com/birkhauser/mathematics/book/978-0-81...
Now, there are arguments for positions like Knapp's, but they're more complicated and contested than Knapp's facile post suggests. See, for instance, this Stanford Encyclopedia of Philosophy article: http://plato.stanford.edu/entries/fictionalism-mathematics/
Knapp latches onto a lack of empirical evidence for a Platonist viewpoint (though narrowly defines evidence as apparently needing to be extra-human to be valid, which isn't a well founded definition, but, then again, definitions are the fundamental problem here), and so he concludes that it must therefore be false, completely missing his own point.
Sorry. While there is a wealth of interesting discussion to be had on philosophy of mathematics, and even more discussion (and certainly more concrete discussion) to be had on the difference between how mathematics is actually created and how it is then defined in proofs and literature, this passage misses the boat.
Also, spitx: while fair use covers quoting from and discussing articles quite well, quoting the full article, even with discussion, could be problematic and certainly should be considered poor form. Take some quotes and then link to the rest.
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.
And what is wrong with saying that? The burden of proof is on Platonists, just like it is on theists. I don't see any reason not to think that Platonism has been a huge waste of time.
Many mathematical areas initially seem to be blind alleys - Hardy rejoiced in the idea that his work on primes would never be a weapon of war. But there's an uncanny tendency for connections to emerge, and theorems in one field imply something new in a different one. Again, it seems too much for mere coincidence.
Moving from what's widely supported to my own views, my suspicion is that reality itself is a mathematical phenomenon (at one time I was expecting to work on the Causal Sets hypothesis, which attempts to explain the universe as arising from certain partially ordered sets).
What about RSA encryption, and the former US export controls thereon?
Also, what about more exotic mathematical objects, like finite fields for example, that have a lot of structural properties but don't seem to model much of anything in the real world, but are still useful theoretically?
That's exactly the point.
>Also, what about more exotic mathematical objects, like finite fields for example, that have a lot of structural properties but don't seem to model much of anything in the real world, but are still useful theoretically?
If that theoretical use applies to physics then to my mind the theoretical objects are as real as anything else. I have a friend who likes to say that electrons don't "really" exist (because they can't be directly observed), they're just a theoretical object that makes our calculations easier. I guess if you pressed me on philosophy I'd say that the only "real" things are sensory observations - we posit objects such as "that table" because they make the explanation of our visual perceptions easier. If positing a finite field makes some part of physics simpler, that seems to be the same thing.
Now for areas of mathematics that are isolated from the rest it's much easier to say they don't represent anything real. But one of the things I was trying to say in the grandparent post is that there's an uncanny tendency for these areas to not stay isolated for long.
To some extent, in CS we study completely arbitrary constructs--Turing machines. Turing machines, indeed, are just tools: they're a simple model of a computing machine that makes sense just based on existing technology. Or perhaps we study the lambda calculus which, while less arbitrary than a Turing machine, is still obviously a tool created by a mathematician to perform a particular task. Or maybe we study existing programming languages which are even more arbitrary and contrived than Turing machines.
All these models? Obviously just tools created by people, often for very specific tasks.
But then a surprising fact emerges: no matter which tool we use, we can accomplish exactly the same things! The set of programs we can run is the same--these arbitrary models are all Turing-complete. Somehow, the notion of Turing completeness is extremely robust against all the different models of computation we've come up with.
In fact, it turns out that any "reasonable" model is going to have the same power. And in this case, "reasonable" actually is reasonable--the main restriction is that the model can only do a finite amount of work at each step.
So while any given model is arbitrary, the class of problems they can solve isn't. Rather, this class naturally emerges from the models. It's a natural property of computation rather than merely a human contrivance.
A similar thing happens when you look at programming and logic. We all know about Curry-Howard: it turns out that writing a program and writing a proof are analogous actions. You can map proof from a logic system to a program in a programming language and vice versa. While both logic systems and programming languages are somewhat arbitrary themselves, the fact that they're somehow equivalent hints at some underlying principle.
And that is why I'm fascinated by category theory, which seems to embody this underlying principle. It unifies proofs and programs with categories in what is very much like an extension of Curry-Howard. But then it goes further, and uses exactly the same model to talk about a whole bunch of different things: set theory, topology and even theoretical physics! (And, doubtless, countless more I'm simply unaware of.) While any of these fields are again just tools--including categories themselves--it seems that they all reflect some deep underlying structure.
Now, of course, I haven't addressed a very important point: is this "underlying structure" a reflection of the universe or just a reflection of the human mind? Is it perhaps a reflection of the underlying structure of reasoning itself?
Coming up with a reasonable answer to any of these questions is going to take a lot more thought and a lot more writing, but I lean towards the latter. The underlying ideas--the minimal starting points that are enough to uncover the structures I talked about--are far too simple to just be an artifact of the human mind which is itself wantonly complex.
So perhaps I am a Platonist myself. I've never thought about it in those terms, but it seems broadly consistent with my general ideas about life, the universe and everything.
> they're a simple model of a computing machine that makes sense just based on existing technology
Turing machines were invented before any working (almost-) universal computer existed.
Category theory arises from probably the simplest imaginable notions of "transformation". There's no way to not critique me for being anthropocentric, but it's hard to imagine an alternative theory where this doesn't exist.
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html