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Language: en
do you think math is discovered or
invented so we were talking about the
different kind of mathematics that could
be developed by the alien species the
implied question is is yeah it's math
discovered or invented is you know it's
fundamentally everybody going to
discover the same principles of
mathematics so the way I think about it
and everyone thinks about it differently
but here's my take
I think there's a cycle at play where
you discover things about the universe
that tell you what math will be useful
and that math itself is invented in a
sense but of all the possible maths that
you could have invented its discoveries
about the world that tell you which ones
are so like a good example here is the
Pythagorean theorem when you look at
this do you think of that as a
definition or do you think of that as a
discovery from the historical
perspective Ray's a discovery because
there were but that's probably because
they were using physical object to build
their intuition and from that intuition
came the mathematics so the mathematics
person is some abstract world detached
from physics but I think more and more
math has become detached from you know
we wouldn't even look at modern physics
from string theory so even general
relativity I mean all math behind the
20th and 21st century physics kind of
takes a brisk walk outside of what our
mind can actually even comprehend in
multiple dimensions for example anything
beyond three dimensions maybe four
dimensions know how your dimensions can
be highly highly applicable I think this
is a common misinterpretation the if
you're asking questions about like a
five dimensional manifold that the only
way that that's connected to the
physical world is if the physical world
is itself a five dimensional manifold or
includes them wait wait wait a minute
wait a minute
you're telling me you can imagine a five
dimensional manifold no no that's not
what I said I I'm I would make the claim
that it is useful to a three dimensional
physical universe despite itself not
being three-dimensional so it's useful
meaningful even understand a three
dimensional world would be useful to
have five dimensional manifold
yes absolutely because of state spaces
but you're saying there in some in some
deep way for us humans it does it does
always come back to that
three-dimensional world for the useful
usefulness at the dimensional world and
therefore it starts with a discovery but
then we invent the mathematics that
helps us make sense of the discovery in
a sense yes I mean just to jump off of
the Pythagorean theorem it feels like a
discovery you've got these beautiful
geometric proofs where you've got
squares and you're modifying there is it
feels like a discovery if you look at
how we formalize the idea of 2d space as
being r2 right all pairs of real numbers
and how we define a metric on it and
define distance no wait hang on a second
we've defined distance so that the
Pythagorean theorem is true so that
suddenly it doesn't feel that great but
I think what's going on is the thing
that informed us what metric to put on
r2 to put on our abstract representation
of 2d space came from physical
observations and the thing is there's
other metrics you could have put on it
we could have consistent math with other
notions of distance it's just that those
pieces of math wouldn't be applicable to
the physical world that we study because
they're not the ones where the
Pythagorean theorem holds so we have a
discovery a genuine bona fide discovery
that informed the invention the
invention of an abstract representation
of 2d space that we call r2 and things
like that and then from there you just
study r2 is an abstract thing that
brings about more ideas and inventions
and mysteries which themselves might
yield discoveries those discoveries
might give you insight as to what else
would be useful to invent and it kind of
feeds on itself that way that's how I
think about it so it's not an either/or
it's not that math is one of these or
it's one of the others at different
times it's playing a different role so
then let me ask the the Richard Fineman
question then along that thread is what
do you think is a difference between
physics and math there's a giant overlap
there's a kind of intuition that
physicists have about the world that's
perhaps outside of mathematics it's this
mysterious art that they seem to possess
we humans generally possess and then
there's the beautiful rigor of mathema
Erick's that allows you to mean just
like as we were saying invent frameworks
of understanding our physical wall so
what do you think is the difference
there and how big is it well I think of
math as being the study of like
abstractions over patterns and pure
patterns in logic and then physics is
obviously grounded in a desire to
understand the world that we live in
yeah I think you're going to get very
different answers when you talk to
different mathematicians because there's
a wide diversity and types of
mathematicians there are some who are
motivated very much by pure puzzles they
might be turned on by things like
combinatorics and they just love the
idea of building up a set of
problem-solving tools applying to pure
patterns right there are some who are
very physically motivated who who tried
to invent new math or discover math in
veins that they know will have
applications to physics or sometimes
computer science and that's what drives
them all right like chaos theory is a
good example of something that it's pure
math that's purely mathematical a lot of
the statements being made but it's
heavily motivated by specific
applications to largely physics and then
you have a type of mathematician who
just loves abstraction they just love
pulling into the more and more abstract
things the things that feel powerful
these are the ones that initially
invented like topology and then later on
get really into category theory and go
on about like infinite categories and
whatnot these are the ones that love to
have a system that can describe truths
about as many things as possible right
people from those three different veins
of motivation and demand are going to
give you very different answers about
what the relation at play here is
because someone like Vladimir Arnold who
is this he's written a lot of great
books many about like differential
equations and such he would say math is
a branch of physics that's how he would
think about it and of course he was
studying like differential equations
related things because that is the
motivator behind the study of PDEs and
things like that but you'll have others
who like especially the category
theorists who aren't really thinking
about physics necessarily it's all about
abstraction and the power of generality
and it's more of a happy coincidence
that that ends up being useful for
understanding the world we live in and
then you can get into
why is that the case that's sort of
surprising that that which is about pure
puzzles and abstraction also happens to
describe the very fundamentals of quarks
and everything else so what do you think
the fundamentals of quarks and and the
nature of reality is so compressible and
too clean beautiful equations that are
for the most part simple relatively
speaking a lot simpler than they could
be so you have we mentioned somebody
like Stephen Wolfram who thinks that
sort of there's incredibly simple rules
underlying our reality but it can create
arbitrary complexity but there is simple
equations what I'm asking a million
questions that nobody knows the answer
to but no idea why is it simple I it
could be the case that there's like a
filter iteration I played the only
things that physicists find interesting
other ones little simple enough they
could describe it mathematically but as
soon as it's a sufficiently complex
system like now that's outside the realm
of physics that's biology or whatever
have you and of course that's true right
you know maybe there's something words
like of course there will always be some
thing that is simple when you wash away
the like non important parts of whatever
it is that you're studying just unlike
an information theory standpoint there
might be some like you you get to the
lowest information component of it but I
don't know it maybe I'm just having a
really hard time conceiving of what it
would even mean for the fundamental laws
to be like intrinsically complicated
like some some set of equations that you
can't decouple from each other well no
it could be it could be that it's sort
of we take for granted that they're the
the laws of physics for example are for
the most part the same everywhere or
something like that right as opposed to
the sort of an alternative could be that
the rules under which are the world
operates is different everywhere it's
like a like a deeply distributed system
or just everything is just chaos like
not not in a string
definition of caste but meeting like
just it's impossible for equations to
capture for to explicitly model the
world as cleanly as the physical does
any we're almost taking it for granted
that we can describe we can have an
equation for gravity mm-hm
for action at a distance we can have
equations for some of these basic ways
the planets moving just the low level at
the atomic scale all the materials
operate at the high scale how black
holes operate but it doesn't it it seems
like it could be there's infinite other
possibilities where none of it could be
compressible into such equations so it
just seems beautiful it's also weird
probably to the point you're making that
it's very pleasant that this is true for
our minds right so it might be that our
minds are biased to just be looking at
the parts of the universe that are
compressible and then we can publish
papers on and have nice e equals
mc-squared equations right well I wonder
would such a world with uncompressible
laws allow for the kind of beings that
can think about the kind of questions
that you're asking that's true
right like an anthropic principle coming
into play it's some weird way here I
don't know like I don't know what I'm
talking about it or maybe the universe
is actually not so compressible but the
way our brain the the way our brain
evolved were only able to perceive the
compressible parts I mean we are so this
is a sort of Chomsky argument we are
just the sentence of apes over like
really limited biological systems so it
totally makes sense there were really
limited little computers calculators
that are able to perceive certain kinds
of things in the actual world is much
more complicated well but we can we can
do pretty awesome things right like we
can fly spaceships and that we have to
have some connection of reality to be
able to take our potentially
oversimplified models of the world but
then actually twist the world to our
will based on it so we have certain
reality checks that like physics isn't
too far afield simply based on what we
can do and the fact that we can fly is
pretty good it's great
yes like and lambic on such a bit the
laws were working with our are working
well
you