
The limited, and hence reasonable, effectiveness of mathematics in physics - discreteevent
http://arxiv.org/abs/1506.03733
======
lisper
"the contents of mathematics is far from arbitrary -- while an infinite number
of mathematical objects might potentially be envoked [sic], the few that prove
interesting develop a very small number of core concepts. These core concepts
are not arbitrary -- they are elaborations of structures which are discovered
during the study of nature. There are four of these core concepts: number,
geometry, algebra and logic."

It could be as well argued that there is only one core concept: computation.
(Or maybe "symbol".)

The interesting question is: of all the myriad possible things that are
computable, how is it that we are able find the unfathomably small subset
whose behavior corresponds to (what we perceive as) physical reality? That was
the question Wigner was asking, and I don't think Smolin answers it at all.

~~~
d4rkph1b3r
"the contents of mathematics is far from arbitrary -- while an infinite number
of mathematical objects might potentially be envoked [sic], the few that prove
interesting develop a very small number of core concepts. These core concepts
are not arbitrary -- they are elaborations of structures which are discovered
during the study of nature. _There are four of these core concepts: number,
geometry, algebra and logic._ "

The fact that Smolin thinks these are distinct things shows just how out of
touch he is, and is ironic given his hatred of mathematics.

~~~
DougMerritt
> ironic given his hatred of mathematics.

Wait, what? Are you going off of a particular polemic of his?

Because it's hard to imagine someone who had to have learned enough math to do
string theory, to be a true math hater, even if (I am assuming) he rants.

------
themgt
This is excellent. Interview going over more here:
[https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-
an...](https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-and-the-
status-of-modern-physics/)

"As Roberto Mangabeira Unger and I argue in our new book The Singular
Universe, the most important discovery cosmologists have made is that the
universe has a history. We argue this has to be extended to the laws
themselves. Biology became science when the question switched from listing the
species to the dynamical question of how species evolve. Fundamental physics
and cosmology have to transform themselves from a search for timeless laws and
symmetries to the investigation of hypotheses about how laws evolve."

and

"Neurosciences are a fabulous area to work in, ripe for great discoveries.
I’ve always felt this and indeed the only alternative to a career in physics
that ever attracted me was a brief flirtation in college with neuroscience.
But that is a field which is as bedeviled by outdated metaphysical baggage as
physics is. In particular, the antiquated idea that any physical system that
responds to and processes information is isomorphic to a digital programmable
computer is holding back progress."

~~~
danbruc
When you listen to some contemporary physicists like Nima Arkani-Hamed they
seem to disagree, i.e. they believe that neither time nor space are
fundamental properties of our universe but instead emerge from something more
fundamental. [1] This leads of course to some really hard problems and
especially it is no longer clear what physics is even about. It used to be the
science of things moving through space over time, but what if you lose space
and time?

[1] [http://www.cornell.edu/video/nima-arkani-hamed-spacetime-
is-...](http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-doomed)

~~~
DougMerritt
> they believe that neither time nor space are fundamental properties of our
> universe but instead emerge from something more fundamental.

Yes, this has gained increasing acceptance ever since its first (?) suggestion
from the late illustrious John Archibald Wheeler in the 1960s, and it is a
very compelling idea, although yes, it makes it hard to conceive what we may
be talking about.

> like Nima Arkani-Hamed they seem to disagree,

I missed it, who's disagreeing with whom about what? The parent comment
doesn't _seem_ to have content disagreeing, unless I'm missing something.

------
skybrian
"As we do not believe in timeless Platonic realities, we do not want to say
that chess always existed-in our view of the world, chess came into existence
at the moment the rules were codified."

Okay, but when were the variations of chess invented? For example, instead of
playing on an 8x8 board, you could play on an x*y board for any natural
numbers x and y. Do we need to set x and y to specific values for that
variation to "come into existence". Does it make sense for us to say that a
game exists when it's never been played?

How specific do we need to be about imagining a game? If we just say that
chess can vary in many possible ways by changing any of its rules, does
checkers come into existence? Or suppose we write a computer program to
generate variations on board games. Do I have to run the program for these
games to "come into existence", or just write it, or just think about writing
it? If the program has a bug so that it won't generate a particular board
game, does that game come into existence when I document the bug, fix the bug,
or verify the fix?

I don't think any philosophy is all that good at distinguishing between
explored and unexplored concepts in the general case. But that's a problem
mostly when you try to do philosophy. If you have a practical purpose in mind,
it's much easier to come up with a useful working definition for the
particular domain you're interested in.

~~~
danbruc
What you are struggling with here is the definition of existence. And
Platonism has this problem in general, you rarely get a sharp definition of
what it means that something exists. Natural numbers exist. I know what
natural numbers refers to but I have no idea what existence refers to. When we
talk about things in the physical world I can come up with a definition of
existence, say anything I can - at least in principle - interact with. This
definition is far from perfect and has quite a few issues itself but it is at
least a start. But what does the existence of mathematical objects even mean?

~~~
DougMerritt
> but I have no idea what existence refers to.

A lot of brilliant famous mathematicians in history have been befuddled on
that point as well.

I think it's a linguistic issue, which perhaps you are hinting at.

If I say "that rock exists", in the real world, that meaning of "exists" is
parallel to, yet _different_ than if I say "prime numbers exist".

It's hardly the first time that one word meant more than one thing.

The linguistic "exists" is a "deictic" \-- I can point to the thing.

The mathematical "exists" means "come, let us reason together, and I can
demonstrate to you abstractly that this one thing is inescapably implied by
this starting point".

(Phrasing borrowed from Leibniz's commentary on Calculus Ratiocinator)

They both have a similar final state, I suppose: that we are convinced of the
"existence" of a thing, but nonetheless the existence in each case is in a
separate realm.

~~~
ttctciyf
I like to consider the mathematical use of 'exists' as something quite similar
to its use in sentences like "a solution to such-and-such problem exists" \-
basically, an assertion that if you do X then Y will result - not so
dissimilar from your "come let us reason together..." explanation, perhaps.

------
jonahx
Anyone interested in this might also like Lakoff's "Where Mathematics Comes
From," which I read after seeing a Bret Victor recommendation of it:

[http://www.amazon.com/Where-Mathematics-Come-From-
Embodied/d...](http://www.amazon.com/Where-Mathematics-Come-From-
Embodied/dp/0465037712)

Absolutely fascinating book.

~~~
ubernostrum
See also Martin Gardner's defenses of mathematical realism for counterpoint.

------
animefan
I think that there are softer versions of Platonism (which in other contexts
would not be considered Platonism at all, but are closer to Platonism that
Smolin's idea) that are somewhat consistent with naturalism. The softer
version is that there _appear_ to be mathematical objects with independent
existence, even though there is no a priori philosophical reason to believe in
their independent existence, and there are indeed reasons to doubt it (e.g.
non-standard models of not only the reals, but also the natural numbers).

One reason I find it hard to reject Platonism entirely is that the main
alternative in the philosophy of mathematics, Formalism, looks to me like a
kind of Platonism. This is because formalism replaces belief in the
independent existence of mathematical objects (real numbers, integers) with
belief in the independent existence of meta-mathematical objects (theorems,
proofs) which most formalists conceive of in mathematical terms.

Ultimately I think that there is no a priori limit that we can place on how
science is going to pan out, and how this will affect our philosophy. E.g. I
find the idea of multiple universes distasteful and nonsensical (e.g. If
everything possible happens, then how can we interpret probabilities? Why do
we live in a universe where macro observations are consistent with applying
the law of large numbers to quantum randomness? I can't think of any
anthropic/observer based principle that would enforce this) but I can't yet
rule out that that's really how things are.

------
amelius
> There is an objective distinction between past, present, and future.

How does that work with relativity, which says that the concept of
simultaneity does not even exist in the sense that we understand it? [1]

[1]
[https://en.wikipedia.org/wiki/Simultaneity](https://en.wikipedia.org/wiki/Simultaneity)

~~~
jerf
There's an objective distinction between "past", "future", and "undefined".
("Present" is sort of more a human concept than a physics one.) Past is in
your past light cone, future is in your future light cone, and "undefined" is
everything else.

~~~
gdavisson
I prefer to categorize them as "absolute past", "absolute present", and
"absolute elsewhere". This is because for any event in one of your light cones
(past or future), there'll be a reference frame that places it at your
location (i.e. the same spatial coordinates, just at a different time). But
for events outside your light cones, no reference frame will have it at the
same location; it's absolutely elsewhere.

BTW, if you also want to talk about the present, you can only talk about it at
a particular location. If you want, you can define 3 "thin" boundary regions:
the surface of the future light cone (boundary between absolute future &
absolute elsewhere), the surface of the past light cone (similar), and the
"here & now" (the single point in spacetime where all 3 regions meet).

------
im3w1l
I have heard of two paradigms humanity have used to try to explain the world

Mathematics/logics is one of them. It says that the world is guided by
_principles_. "Objects contain different amount of phlogiston and when they
burn it is released."

Anthropomorphism is the other. I.e. there is one, or many human-like beings
with personalities and feelings. And things that happen happen because the
beings want them to. "If you offend the fire spirits your house may burn down"

Are there more?

~~~
xioxox
Many things happen the way they do just down to luck or chance. If that
microscopic quantum fluctuation hadn't been there our galaxy wouldn't have
existed.

Some people believe in karma. Maybe this is covered as a principle, however,
even though it isn't scientific.

Perhaps solipsism might also count as an "explanation" of the world. I exist,
so that explains the world.

Fate is also an explanation for what happens.

