Earlier quoted context omitted.
Here's an amusing counterargument: http://pauli.uni-muenster.de/~munsteg/arnold.html
That's not an argument, that's an almost incoherent rant.
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The Paradox of the Proof
71–80 of 124 posts
Re: The Paradox of the Proof
#72If a programmer locked himself away for 14 years and then emerged and announced he'd written a completely bug free OS, there would be skepticism. Code needs to be battle tested by other people to find the bugs. 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…
Quote from original article
"...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.
Re: The Paradox of the Proof
#73This is a great article. The writer has obviously spent a lot of time speaking to Mochizuki’s colleagues, and has explained the whole strange situation in a way that is layman-friendly without being wrong. It’s interesting that it’s part of an experiment in donation-funded journalism: I was sufficiently impressed that I donated a few dollars, but I fear that is not likely to be a common enough response to be a viable…
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…
The only problem I see with this is that value isn't always proportional to time. Sometimes I spend a long time reading an article because it is written in an inscrutable style and it takes a long time to see what the author is saying. Other authors write so well that I can follow their arguments as fast as I can read. All other things being equal, the later article is more valuable, but I spend more time on the former article.
Re: The Paradox of the Proof
#74As 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 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!
Re: The Paradox of the Proof
#75"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…
The book that you want to read is The Mathematical Experience . Trust me on this. There is no book that I know which is as good at conveying the feeling or breadth of doing mathematics, with real examples, at a level that is both largely accessible to high school students, and enlightening for math professors.
Thanks for the suggestion. Sounds like an interesting read.
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...
Re: The Paradox of the Proof
#76> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. 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 coincid…
This is a philosophical position that almost no one engaged in serious math holds.
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.
Re: The Paradox of the Proof
#77If a programmer locked himself away for 14 years and then emerged and announced he'd written a completely bug free OS, there would be skepticism. Code needs to be battle tested by other people to find the bugs. 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…
Isn't one guy working alone for 6 years how Fermat's Last Theorem got solved? It did have a flaw that got fixed later in collaboration, but most of the work was one individual's deep focus.
Wiles was much more involved in the Math community at large than Mochizuki. He continued to publish non-FLT papers, go to conferences, etc. Even then, the general consensus is that FLT would have been proven much faster if Wiles shared his work earlier.
Re: The Paradox of the Proof
#78As 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…
This really deserves a longer and better answer, but I'm struggling to explain. It's a pretty deep conceptual thing, but I'll try.
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.
Re: The Paradox of the Proof
#79> For centuries, mathematicians have strived towards a single goal: to understand how the universe works, and describe it. 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 coincid…
I think there's a reasonable argument to be made that mathematical constructs are part of the universe. 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…
"I think there's a reasonable argument to be made that mathematical constructs are part of the universe."
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.
Re: The Paradox of the Proof
#80“The point is not to prove the theorem,” explains Ellenberg. “The point is to understand how the universe works and what the hell is going on.”
School failed at conveying this to me in so many domains.