

It’s science if it bites back - gnosis
http://scottaaronson.com/blog/?p=29

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michael_dorfman
_That 17 is prime strikes us as absolutely certain, yet there’s nothing in the
physical world we can point to as the source of that certainty. (Seventeen
blocks that can’t be arranged into a rectangle? Give me a break.) In that
respect, math seems more like subjective experience than science: you might be
wrong about the sky being blue, but you can’t be wrong about your seeing it as
blue._

Here's where a little philosophy would help; we could save a lot of time by
invoking the sensible/intelligible distinction. Mathematical entities are
intelligible, not sensible, and their mapping to the external world is
epiphenomenal. Seventeen is prime, not because of any empirical evidence, but
because it _has_ to be prime under the definition of primality. Math, like
chess, is a game with rules.

~~~
shasta
Did that really help? On my screen, your paragraph is about 10 characters
longer than his, and about twice as opaque.

> Seventeen is prime, not because of any empirical evidence, but because it
> has to be prime under the definition of primality.

This is rather question begging. There are plenty of sets of rules that are
inconsistent. The real question is why following the rules of
arithmetic/primality never seems to lead to contradiction. There are mainly
two reasons why we believe these rules to be consistent: we've used the rules
over and over and have never found any contradiction, and we perceive the
rules to be about something - the natural numbers. If you perceive the natural
numbers the way that I do, then it should seem clear to you that searching for
a contradiction in arithmetic is a fool's errand. So are these natural numbers
that we perceive "real"? How do we perceive them? My opinion is that
philosophy of such matters is a rather hopeless endeavor. I'm sure the
philosophers have lots of new definitions and concepts built up to describe
the situation, but how much progress can you really hope to make here?

~~~
michael_dorfman
_Did that really help? On my screen, your paragraph is about 10 characters
longer than his, and about twice as opaque._

It's only opaque if you haven't read Kant; if I am going to have to explain
Kant to you, it's going to take a hell of a lot more characters than the extra
10.

 _There are plenty of sets of rules that are inconsistent. The real question
is why following the rules of arithmetic/primality never seems to lead to
contradiction._

Why is _that_ the question? I don't know of any mathematicians particularly
interested in that question at all, actually.

 _So are these natural numbers that we perceive "real"? How do we perceive
them?_

Not, _that_ is a more interesting question. My answer is already implied by
what I wrote above (in Kantian terms). We don't _perceive_ natural numbers
with the senses. A number is not a physical thing in the world-- it is an
_abstraction_. We've also invented a series of manipulations we can perform on
these abstractions, much like we have invented the rules of chess. And, there
are a lot of extremely interesting things that come out of our manipulating
the abstractions according the the rules we've set up.

However: asking why the number 17 is prime, is like asking why a bishop can
only move diagonally. The bishop moves diagonally because _that's what a
bishop does_. Seventeen is prime because _that's what prime means._ We don't
need recourse to the external world of empirical sense perception to answer
these questions.

Was that less opaque?

~~~
shasta
(First, apologies for use of "opaque" which was needlessly inflammatory.) I've
read "A critique of pure reason" some years ago, but honestly I found much of
it to be rather dodgy. Take this:

>We don't perceive natural numbers with the senses. A number is not a physical
thing in the world-- it is an abstraction. We've also invented a series of
manipulations we can perform on these abstractions, much like we have invented
the rules of chess. And, there are a lot of extremely interesting things that
come out of our manipulating the abstractions according the the rules we've
set up.

I see this as new words describing the situation, but very little progress.
Are these abstractions "real"? How do we perceive them? I agree it's not with
our eyes and ears.

> However: asking why the number 17 is prime, is like asking why a bishop can
> only move diagonally. The bishop moves diagonally because that's what a
> bishop does.

"That's just the way it is" isn't an explanation of anything, though. How do
we know, once we've proven 17 prime, using these rules, that we will never
discover a factorization of 17? How do we know that our reasoning about these
abstractions is sound?

~~~
michael_dorfman
_Are these abstractions "real"?_

That depends, of course, on how you define "real". Are the rules of chess
"real"?

 _How do we perceive them? I agree it's not with our eyes and ears._

And here we're back to Kant. They are _intelligible_ , not _sensible_. We
perceive them with our mind/intellect. (Interestingly, in Buddhist philosophy,
there are six senses-- the mind joining the other five, so they push the
intelligible/sensible distinction down a level.)

 _"That's just the way it is" isn't an explanation of anything, though. How do
we know, once we've proven 17 prime, using these rules, that we will never
discover a factorization of 17? How do we know that our reasoning about these
abstractions is sound?_

I'm not saying _"That's just the way it is"_ \-- I'm saying "That's the way it
_has_ to be because we defined it that way." How do we know that in some
particular game of chess, the bishop won't suddenly be allowed to move
sideways?

Anyway: if you are really interested in the reduction of arithmetic to pure
logic, check out Russell and Whitehead's Principia Mathematica-- or better
yet, read the graphic novel "Logicomix" which does a wonderful job of covering
the domain for non-specialists.

------
fragmede
> Seventeen blocks that can’t be arranged into a rectangle? Give me a break.

Give _me_ a break.

Is multiplication anything but shorthand for counting the number of blocks in
an M by N rectangle?

So then, how is that not an adequate whatever for 17 being prime? It may not
be be the traditional math you learned in school alongside long division and
multiplication, but it's math all the same.

~~~
Confusion

      Is multiplication anything but shorthand for counting the
      number of blocks in an M by N rectangle?
    

Originally, it probably was (but dive into the history of maths to be sure).
After various abstractions and generalizations, it isn't anymore.

~~~
arethuza
Multiplication in a finite field is quite straightforward, clearly not based
on the area of a rectangle and highly relevant to CS.

------
Groxx
They seem surprised that mathematicians, who deal very strictly in almost pure
logic, ply their skills in the philosophical realms. I highly doubt this has
anything to do with math's "mystical tendencies" and more to do with similar
skill sets.

Also, apparently they haven't taken even moderately high level math courses,
which give much more concrete definitions of divisibility. 17 is prime because
it matches the _definition_ of "prime". It also _doesn't_ match the definition
of "even". What of it? Blue is blue because it's blue, not because it can't be
arranged into a rectangle.

