Thoughts about big questions
Back to `Random and Silly'
Oliver Knill
Thinking about fundamental questions should not be limited
to ``important people". Their point of views are valuable
but often skewed due to extraordinary circumstances. So, here is
the point of view of an ``average Joe". I don't
really think it matters much what any individual person thinks
as it usually is all wrong anyway. History shows that even
the smartest folks are often incorrect in basic questions or
predictions. History also shows that everybody always overestimate topics
related to their own little world. Still, ``big questions" should not
be ``off limits" and everybody should try or be allowed to answer
such questions from time to time even so the result is almost certainly
futile or awkward. Better spent time is to try to answer question
which are interesting. So, this is really short:
- Is math invented or discovered?
This is not such an interesting question as we have no
way to decide this. Both scenarios are possible and
we have no tools to figure out which is right. It also
depends on the precise meaning what we think to be the "inventor".
A good answer was once given by Kontsevich at a panel discussion:
"good math is discovered, while lots of invented math should be
forgotten". Why can the question not be answered?
If we were part of an elaborate
computer simulation the rules would have been invented by
a game designer. On the other hand if it would have be
invented by us only, then it would not exist without us, which is silly too. Is
the Pythagorean theorem gone, if humanity would get blown away by a solar storm? Or did it
not exist 10 thousand years ago? On the other hand, that theorem could have been designed
by a game designer and that architect would also have
created the proof system allowing us to give a proof. In some sense
we have wiggled us out of this system by creating non-Euclidean,
Riemannian and non-commutative geometries already where the theorem needs
to be modified, but these additional layers could all have been created too
by the game designer to contain any ``smarty pant" trying to escape the
game. This leads to:
- Do we live in a computer simulation?
We have not noticed a glitch in the matrix yet!
It would be depressive, if we would live in a game
because you know that the kid playing it in a console could switch
it off at any time. Camera zoom to a level up in the simulation:
"Johnny! Dinner time!" End of game.
You can imagine the game designer to have built nicely
looking theories like string theory and laugh hard
by watching humans follow the clues and see them get to
nothing. We already have computer games which make one
really believe to live in an other world. For me personally,
this has happened already in primitive games like old Atari
games or Myst or one person shooters. For modern games it is
even more extreme. We can today
explore Alexandria
as it was 2000 years ago.
- Is math a tool for science?
It is maybe more like that science is a playground for math.
Math is widely seen as the universal language for all sciences but it
is more: essentially all mathematical structures
are also realized in nature somewhere. I used to tease
my high school friends following biological sciences
that physics is part of math, chemistry is part o
physics, biology is part of chemistry, medicine is part
of biology so that all is just math ...
There is a famous XKCD cartoon which brings it to the point.
- Is there a god?
It depends on the definition. If it is a concept like mathematics, yes,
if it is an old man or women sitting in a cloud surrounded by
angels, then I don't think so, even so we can not
exclude it. It would be silly although. Also non-existence proofs
are not convincing: "God is almighty. So, it
can build a wall it can not climb, contradicting the
property almighty."
On the other hand, it is hard to imagine that just
all is random. Completely random structures are not
that nice. The fact that there exists interesting
mathematical structures, allowing for complex systems
to evolve is amazing. We could imagine a frame work, where
every axiom system is inconsistent, not allowing to do
any useful science at all. There are models in statistical
mechanics which explain the emergence of nice structures
from chaos. It is the emergence of
language and logic which allows us to explore such
pattern formations. Maybe the best answer to the question is
given by Pantheism which is the point of view that
everything including us is god. A measurable justification
is that this word is one of the most common words used as passwords...
This leads to a much better question:
- What programming language is the best
Every language has its use. I personally like both high level
programming languages like computer algebra systems as well
as very low level programming languages like Turing machines.
I myself program mostly in scripting languages
(Perl, Bash, Javascript, Python) and computer algebra
(Mathematica Example, Pari Example) and specialized systems (Povray,
OpenCas, Processing), sometimes in fast compiled languages
like C [Example: Chirikov, or
Omnidirectional vision]
or Pascal [Example: almost
periodic sphere packings]. And then there are lots of languages
that are border-line programming languages (even so
many are Turing complete) like LaTeX, WebGL, Php, AIML, HTML, XML, SVG
or Postscript in which it can be good to be fluent in.
Additionally, it is useful to know well a few
basic data formats like for picture, movie, music, styles.
More important is the ability to translate between formats and if
necessary to be able write a translator or driver.
Sometimes it is useful for example to write down by hand
each pixels of a picture as a .PPM file (containing triples of
red,gree, blue values) [I did that for example for
these pictures as it gives control about each pixel] or of reach triangle of an
object as an .STL file (containing triples of numbers as well as
normal vectors) with to be 3D printed.
I often write programs which write programs for other languages.
An example is Mathematica writing a javascript procedure which
can be watched with a browser [Example]
or a file which can be run through a ray tracer [An example]
or Mathematica exporting an OpenScad program which appears in this article
or then probgrams write themselves.
Having recovered again from too deep questions, here is the next:
- What religion is the best?
Pluralism
takes the best of all religions. Especially holidays! Seriously, we don't
have enough holidays. These are the times during which we are most creative.
There is some correlation between religions with lots of holidays and
scientific or artistic successes of that culture. Mathematicians often
prove theorems during holidays (the prototype is the
Christmas theorem
in number theory which was found by Fermat in a letter of December 25th).
So, if anybody will build a religion with the first amendment "You shall have
lots of holidays", then I'm in.
- What method in science works best?
Dogmas and methods are more like limitations. Rather than
a method, we need fact checking tools to assess whether we
make progress at all and not just build on sand.
Any physical theory for example should relate to experiments
(which can be mathematical thought experiments). Having been exposed to
lectures of Paul Feyerabend (who also saw anarchism as a scientific
method) as an undergraduate, I learned also to value the
"no method" approach. But whatever works, works. If "method"
works, let it be the "method". If chaos works, let it be chaos.
What is more important is the ability to measure whether something
"works". And that requires quantitative, rather than qualitative
results.
In mathematics, a theorem which gives equalities or estimates
is much less likely to be false as a result which is not quantitative and
just gives existence or worse, is speculation or hypothesis.
A result claiming that a certain type of maps has three fixed points is
a quantitative statement. It can be tested in examples. Quantitative
statements are more reliable if they are constructive as again, we can
see that things work. Also in physics, results which give quantitative
measurable results are the most valuable. The gravitational wave
measurements are one of the most recent examples.
- Is mathematics consistent?
There should exists an axiom systems which is consistent.
It would be depressive it would all be in vain. If every
axiom systems was eventually inconsistent (meaning we could prove
a false statement in each of them), we would have to retreat to nihilism which
(quote from the movie Lebowski) is ``not even an ethos".
ZFC most likely is. Even more likely is Quine's new foundation. Currently,
Lawvere's elementary theory of the category of sets ETCS looks good
as it is also flexible, allowing for example to work in a
finitism
framework. We know by Goedel, that it is not possible to figure out within a given
axiom system, whether it is consistent. This is similar as that
we can not figure out whether we are part of a computer simulation.
It is turtles all the way round. Even if we would find a glitch in the
matrix, the game designer having programmed it could be part of a simulation
too.
- Are the multiverses?
By definition, the universe is all we can ever
access or which affects something we can ever measure.
In the Wikipedia article
about it, Newton is already credited with the idea but Newton
made the observation that we can imagine different physics. And that is
what happens in mathematics. We can simulate the Newton problem in any space
in any dimension and analyze it, we can look at gauge theories in any space
and dimension with any gauge groups, look at different versions of quantum mechanics
or different string theory landscapes, or using different new mathematical structures.
This is all great as it is math and that was what Newton already pointed out. In physics,
if something should happen in an universe what is inaccessible to us
is just not of any interest to us. ``Inaccessible" of course means that it
has no measurable consequence for us, neither in the past nor in the future.
The problem with the term is not so much the confusion about the definition but
the advocacy to change in the point of view what science is. A much less reliable approach.
The discussions about multiverses could be as pointless
as the questions about whether we live in a simulation or whether the term "manifold"
refers to a connected space or not (it is a pointless discussion as it is a definition).
The topic however makes for some good science fiction or comedy. Maybe the entire thing
is just a plot with the goal to entertain.
Document history:
Started: August 23. 2018
updates and more links added on August 24, 2018