
The Paradox of the Proof - ColinWright
http://projectwordsworth.com/the-paradox-of-the-proof/
======
robinhouston
This 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 way of funding this sort of time-
consuming journalism. I would love to be wrong about that.

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/>

~~~
calinet6
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. 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.

~~~
msilenus
There is a micro-payment service called Flattr [1] which does something like
this. It was founded by Peter Sunde, one of the pirate bay founders.

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.

[1] <https://flattr.com>

~~~
gwern
Someone actually mentioned Flattr to me today as a way to take donations for
my own writings; I've always begged off because I've never wanted to take the
time to set it up and clutter my site (more), but now here I see Flattr
brought up again... Is there any way of even roughly estimating how much
Flattr might bring in?

~~~
ics
> 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 button between PayPal and BTC.

~~~
gwern
> 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.

~~~
qznc
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>

~~~
gwern
Yeah, I just added it to <http://www.gwern.net/> . At first I was all 'o.0 you
expect me to load a bunch of JavaScript just to add a button' but then I
spotted that there was a purely static version and added it without a problem.
We'll see how it goes.

------
pfedor
This reminds me of something I witnessed when I worked at Google. There was
this long-standing problem, I don't want to go into details but it had to do
with websearch indexing and it had gone unsolved for years though it was
regularly affecting search results in a negative way.

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.

~~~
darth_aardvark
As someone newly entering the workforce, this is a really depressing view into
how bureaucracy can squash innovation.

~~~
jerf
Do not be depressed, at least because of this. Few people will ever write
anything that is correct, can not be meaningfully simplified, yet
incomprehensible and essentially unreviewable. The only other thing that comes
to mind are the top-grade encryption algorithms. (They are reviewed, but even
after extensive, extensive review by the very smartest people in the field,
there's still an irreducible part when selecting an algorithm where everyone
still just has to sort of hope there isn't some fatal flaw in there somewhere.
Often, years and years later, there is.)

~~~
xtrumanx
> Do not be depressed, at least because of this. Few people will ever write
> anything that is correct, can not be meaningfully simplified, yet
> incomprehensible and essentially unreviewable.

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.

------
krajzeg
Don't get me wrong, but from what I've seen of the world of science, the only
people who exhibit this type of behaviour are frauds and delusional
pseudoscientists.

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.

~~~
sigmavirus24
Also making claims about typical mathematicians and attempting to apply them
to those already determined to be atypical is a bit dodgy in and of itself.
Perelman was not your typical mathematician and he behaved atypically as well.
I'm not certain but I don't think he travelled and lectured on his proof of
the Poincaré Conjecture. He even turned down a sizable award size. Just
because an atypical scientist behaves atypical doesn't make their work any
less valid.

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?

~~~
krajzeg
I'm far from claiming that the proof is invalid based on my feelings on the
subject.

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.

~~~
sigmavirus24
I must have read the article differently from you. I sensed there was a
positive attitude towards him, but I didn't feel they were trying to say he
had proven the theorem or was so brilliant no one else could grasp it. On the
contrary, it seemed like they were saying "This person has been brilliant for
quite some time, but this 'proof' seems entirely nonsensical even to the
experts in the field."

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.

------
codeulike
If 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 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>)

~~~
pseut
It's a shame that you've disregarded all of the replies to your original
comment. tl;dr there are degrees of bugs and many are easily fixed.

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.

~~~
codeulike
I don't really accept the notion that inconsistencies in a giant mathematical
proof will always show themselves. That does happen sometimes, but if you're
breaking new ground (as Mochizuki seems to be) its much more likely that
things will seem consistent to you, but are actually inconsistent because you
made a mistake somewhere.

~~~
pseut
Right, that's why other people are trying to verify the proof and aren't just
taking it on faith. They're almost certainly going to find errors, the
question is whether or not those errors are easily fixed, difficult to fix, or
fundamentally impossible to fix.

------
brazzy
> 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 coincidence, but not very interesting to
mathematicians - that's what physicists do.

~~~
crntaylor
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 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.

~~~
brazzy
> I think there's a reasonable argument to be made that mathematical
> constructs are part of the universe.

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.

~~~
mcguire
" _By that standard, all fiction is 'a very real part of the universe'._ "

Strangely, I believe that statement supports crntaylor's argument, being the
point of fiction as far as I can see.

------
imdhmd
I actually read the whole article word-to-word. May be this personality of
Shinichi Mochizuki appeals to me or may be i find maths more interesting than
i admit. But i dont know which.

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

~~~
tzs
> 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.

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.

------
pesenti
In case anyone wants to give it a try, here is the paper:
[http://www.kurims.kyoto-u.ac.jp/~motizuki/Invitation%20to%20...](http://www.kurims.kyoto-u.ac.jp/~motizuki/Invitation%20to%20Inter-
universal%20Teichmuller%20Theory%20\(Expanded%20Version\).pdf) and it looks
like he is about to give a lecture in Tokyo:
<http://www.kurims.kyoto-u.ac.jp/~motizuki/news-english.html>

And here is a FAQ on the theory:
[http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-U...](http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-
Universality.pdf)

------
swatkat
Wow! That was an amazing read! These old articles might be of some interest as
well:

[http://today.uconn.edu/blog/2012/10/the-mochizuki-theorem-
wh...](http://today.uconn.edu/blog/2012/10/the-mochizuki-theorem-when-youre-
so-smart-nobody-can-check-your-work/)

[http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-
be...](http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved/)

------
vilda
It's a good tradition in math that you as the author have to convince others
that your theorem is correct. If you cannot convince other fellow colleagues
then you can't claim you have a proof.

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 :)

------
neogodless
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 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!

~~~
ColinWright
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.
    

You're not alone in this, but as you go on in advanced math you find more and
more that multiplication is not really repeated addition, it just happens to
coincide with repeated addition when that makes sense. The problem/opportunity
is that multiplication still makes sense when repeated addition doesn't.

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.

------
pjungwir
This situation reminds me of one of my favorite stories about Plato [1]. He
gave a lecture in Athens on The Good, and lots of people showed up, but the
whole talk was about mathematics! I hope some folks somewhere will take the
time to understand Mochizuki's world. The part about Yamashita studying with
him privately is encouraging.

[1] described in this paper:
[http://www.jstor.org/discover/10.2307/4182081?uid=3739856...](http://www.jstor.org/discover/10.2307/4182081?uid=3739856&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102197047651)

------
adlq
I think this points out the necessity to develop better proof assistant
systems [1], in particular for automated proof checking [2]. However, I have
never interacted with such systems and thus don't know whether it will be
possible to just feed Mochizuki's formidable constructions into it.

[1] <http://en.wikipedia.org/wiki/Proof_assistant>

[2] <http://en.wikipedia.org/wiki/Automated_proof_checking>

~~~
kd0amg
I would not expect so. Mechanized proof systems tend to require a lot more
detail than one would put into a proof meant for humans to read. There's been
a lot of work in automating part of the generation of a proof, but that still
requires a human to look at what the automation came up with and intervene to
guide it in the right direction.

------
cookingrobot
Mochizuki has this 1 page pdf posted on his website that tries to explain
Inter-universal Teichmuller Theory through analogy to a Japanese animation:
[http://www.kurims.kyoto-u.ac.jp/~motizuki/sokkuri-hausu-
link...](http://www.kurims.kyoto-u.ac.jp/~motizuki/sokkuri-hausu-link-
english.pdf)

I wish I could find the animation itself, but that link is broken.

~~~
ics
Song from the album: <http://youtu.be/feOLbipVGEU>

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>).

~~~
stianan
Is that article written by Mochizuki? Can you make sense of it?

~~~
ics
It is not written by Mochizuki (望月新一), but rather by Shiraki Yoshinao (白木善尚).
You can see his profile here:
<http://www.sci.toho-u.ac.jp/is/lab/shiraki_lab/shiraki.html>

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).

------
raverbashing
(ok, rant ahead)

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

<http://en.wikipedia.org/wiki/Abc_conjecture>

~~~
claudius
If you want general results, you need to abstract away from standard preschool
algebra and build up models that then let you get said general results. That
might appear ‘weird’, but I don’t see how else you could get even to calculus,
not to mention, for example, the algebra driving a sensible description of
quantum mechanics or differential geometry.

You called it a rant, but could you maybe still make a suggestion on how to
_better_ think about problems?

~~~
raverbashing
" 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 (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)

~~~
claudius
And after you spent ten years reformulating basic maths in your fancy new
logic, people will look at your papers and won’t understand a word, which
appears to be more or less what happened to our poor protagonist in the OP.

Furthermore, I have to admit I don’t see the immediate advantage such a
reconstruction would bring with it.

~~~
raverbashing
Well, the OPs reinvention looks like something more high level

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>

~~~
claudius
> There are several hard problems in one domain that are trivial in another
> domain, for example.

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.

------
jh3
The first section of this article reminds of the film Proof.

<http://www.imdb.com/title/tt0377107/>

------
tzakrajs
“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.

~~~
ColinWright
The point I often finish with when I give talks is this:

    
    
        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 !!
    

The kids seem to like that ...

------
spitx

      "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 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.
    

This explanation is very satisfying yet disillusioning at the same time.

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...](http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-really-
exist/)

Edit: Clean-up and rewrite.

~~~
voyou
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 only reason the rock works as a tool is
because it has an independent existence (I can hit things with it); the same
might be true of maths.

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/>

~~~
magicalist
Seriously. At least Leah Libresco had the intellectual clarity to declare
herself a Platonist, as in, literally, a believer in or adherent to some part
of Plato's philosophy.

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.

