Fwd: Foundations and Foundationalism

[jodmos.horon]

On Friday, June 24th, 2022 at 04:58, Timothy Y. Chow <tchow at math.princeton.edu> wrote:
> Thousands of philosophers throughout history have convinced themselves that they have
> laid solid foundations once and for all, but come on...who are they kidding? The
> Standard Model of physics is an amazing accomplishment, but once we reach the point
> where disagreements can no longer be settled by experiment, what is there left to do?
> It is bizarre and anomalous that such a high degree of agreement is achievable in
> mathematics."

Well, this observation about physics and mathematics always bring me to the proof of the
irrationality of the golden ratio.

>From a physics point of view, if you're a carpenter with two wooden planks of different
lengths, and you wish to know how many planks of each size you need to juxtapose to bring
them to the same length (a rather natural problem), you are naturally led to perform
euclidean division of the first wooden plank by the second smaller one. You fit n times
the smaller one in the longer one, you the get a remaining piece of wooden plank that
remains after euclidean division. You chop it off, then reproduce the division of the
smallest plank with the chopped off remainder.

It's a rather natural approach, not formal nor formalised, that, as a carpenter, I
believe you are naturally led to. And which implicitly leads you to a Bézout GCD
algorithm this way. And the chopped off remaing chunk gets smaller and smaller. When it
gets to zilch you do get the idea that Bézout's algorithm always ends with 0, a
reasonable observation from an empirical standpoint, and that you can thus find two
integers n and m such n times the long wooden plank is m times the smaller wooden plank.
All lengths in nature are rational.

As a statement about physics, it is not an observation that you may disprove empirically
in any way. Differentiating empirically between 0 and zilch is not possible.

However, once you get a regular pentagon (and better, an exact construction of it), it
is straightforward to check that the diagonals you may draw from it yield a smaller
pentagon within, delimited by the diagonals. And attempting to reproduce Bézout's
algorithm on a diagonal and a side of the greater pentagon brings you immediately to
same problem concerning the smaller pentagon delimited by the diagonals. After two
euclidean divisions.

This observation proves that Bézout's algorithm never may end as one may zoom in the
smaller pentagons ad libitum. And the golden ratio is hence irrational.

This is an example of how a logico-mathematical statement is able to go beyond
empirico-physical investigation to yield a given truth. About a notion that, at least at
the time, was not considered to belong to two distinct scientific domains, mathematics
and physics.

To explain why the empirico-physical method of the carpenter does not succeed in showing
that statement and why the mathematico-logical succeeds, it is not enough to say that
"mathematics is logic or axiomatic" or whatever and then say that physics is a wholly
different matter where the same rules does not apply. Indeed, the carpenter's approach to
fitting wooden planks and the geometer applying euclidean division to lengths of sides
and diagonals of a regular pentagon are rigorously the same processes. There is no
difference in essence concerning the operational methods used to investigate the question
at hand.

What differentiates the logico-mathematical approach from the empirico-physicial one is
the observation that one may exactly reproduce the problem one has with the greater
regular pentagon into the smaller regular pentagon delimited by the diagonals.

What the mathematician then observes is that the problem with the greater pentagon and
the one with the smaller pentagon are rigorously of the same logical strength. What
applies to one applies to the other.

A physicist of an empirical mindset might leave the question open. "The process goes on
from the greater pentagon to the smaller one, and may go on for a long time, but we won't
know if it ends until we get there, to infinity. Which we can't. Hence, we can't know".

Physics is kind of the poster child of the scientific method. Conceived as a process of
conjecture and refutation. Whether it be Popper or Descartes, refutation is key.

Philosophy and mathematics, from antiquity, assert that a different kind of method than
the modern "trust only your senses or instruments" applies. The one where all arguments,
when analogous, must be treated on an equal footing. A no free lunch attitude to truth,
proof and evidence. Whether it be greek mathematics where axioms ended up being the
building blocks of the common rules for truth, whether it be enumerative philosophical
traditions of ancient indian philosophy, where rational enquiry was supposed to proceed
methodically from analysis of concepts with enumerative rules and enumerative procedures
to evaluate, on a prescribed footing, the weight of evidence coming from tradition,
intellect or experience, there is this common assertion of a Method: equal premisses
yield equal conclusions. Or rather, premisses of equal logical strength yield
conclusions of equal logical strength. What applies to one applies to another if
rigourously of equivalent logical strength. If rigorously analogous.

The proof of the irrationality of the golden ration by means of regular pentagons and
their diagonals concludes that the process of euclidean division may not end precisely
BECAUSE equal logical strength is proved. And the conclusion must therefore always be the
same.

In mathematics, contrarily to physics, all exterior phenomenon that may interfere with
such reasoning based on the motto "What applies to one applies to another if rigourously
of equivalent logical strength" have been smoothed out. It's a small bubble, a sandbox
for pure reasoning. It is because it is thus sandboxed that we may observe that this
motto never fails. Or rather, from a popperian viewpoint: the conjecture that this motto
works has never been refuted.

This motto is therefore part of the scientific method as it survived all tests we have
thrown at it. This does not prove it in an absolute sense at all. It just shows that it
has never been falsified when tested in the right, mathematical, conditions. Which is a
good thing: the scientific method is not based only on experimentation, but also needs a
conceptual world of ideas where facts are weighed against one another. If the weighing
procedure was an intrinsically broken or unreliable one, one where logical strength
could be violated, it would be put into question. One of the products of mathematical
activity has been to show that there is no reason to believe that such a procedure for
weighing evidence is a fundamentally broken one. No matter how much a skeptic you are
towards logic itself, you are compelled to yield at one point and concede that weighing
evidence cannot violate logical strength when performed in the right conditions: at
least in mathematics when considered as an experimental science.

> Somehow we're lucky to have one success story, but I see no grounds to be optimistic
> about similar successes in other arenas, where there is no credible mechanism for
> settling disputes definitively.

First step is to expose bad faith. Second step is to convince people that dispute solving
mechanisms matter (a lot of people are "skeptic" about precisely that...) Third step is
not to hope for too much...

(And I'm more worried about the way of settling disagreements in social sciences than I am about solving disputes in theoretical physics, where people seem to have been doing a good job on that count overall...)

Jodmos Horon.