Funny to use the word transcendental there, since the Pythagoreans held ratios to be divine but couldn't figure out irrational numbers, like pi. They had trouble squaring that circle.
That's why I think Pythagoras refused to write down his doctrines. He knew there was more to be empirically discovered -- and he was wary of how text could become dogma. The divine he uncovered was based on a mathematical, harmonious cosmos; but he recognized it was beyond understanding in a lifetime. That's why Pythagorean mysticism is compatible with modern science -- he didn't write anything down!
2000 years later, Kepler had faith in a harmonious cosmos, and charged his model of harmony so it could fit the evidence. He elipsed the circles, instead of squaring them.
Fun fact #1: it is impossible to square a circle [1]
Fun fact #2: the Pythagoreans conducted the first attested scientific experiment in Western history (according to a recent PhD thesis at UMich [2])
Not a mathematician, but it was Thomas Paine's view that the hand of the divine can be seen by the study of nature - which I believe was a view shared by Isaac Newton.
Thomas Paine wrote on his point of view in "The Age of Reason" [0]:
> Each of those churches show certain books, which they call
revelation, or the word of God. The Jews say, that their word of God
was given by God to Moses, face to face; the Christians say, that
their word of God came by divine inspiration: and the Turks say,
that their word of God (the Koran) was brought by an angel from
Heaven. Each of those churches accuse the other of unbelief; and for
my own part, I disbelieve them all.
> As it is necessary to affix right ideas to words, I will, before I
proceed further into the subject, offer some other observations on the
word revelation. Revelation, when applied to religion, means something
communicated immediately from God to man.
> No one will deny or dispute the power of the Almighty to make such
a communication, if he pleases. But admitting, for the sake of a case,
that something has been revealed to a certain person, and not revealed
to any other person, it is revelation to that person only. When he tells
it to a second person, a second to a third, a third to a fourth,
and so on, it ceases to be a revelation to all those persons. It is
revelation to the first person only, and hearsay to every other, and
consequently they are not obliged to believe it.
> It is a contradiction in terms and ideas, to call anything a
revelation that comes to us at second-hand, either verbally or in
writing. Revelation is necessarily limited to the first
communication- after this, it is only an account of something which
that person says was a revelation made to him; and though he may
find himself obliged to believe it, it cannot be incumbent on me to
believe it in the same manner; for it was not a revelation made to me,
and I have only his word for it that it was made to him.
> ...
> But some, perhaps, will say: Are we to have no word of God- no
revelation? I answer, Yes; there is a word of God; there is a
revelation.
> THE WORD OF GOD IS THE CREATION WE BEHOLD and it is in this
word, which no human invention can counterfeit or alter, that God
speaketh universally to man.
> …
> It is only in the CREATION that all our ideas and conceptions of a
word of God can unite. The Creation speaketh an universal language,
independently of human speech or human language, multiplied and
various as they may be. It is an ever-existing original, which every
man can read. It cannot be forged; it cannot be counterfeited; it
cannot be lost; it cannot be altered; it cannot be suppressed. It does
not depend upon the will of man whether it shall be published or
not; it publishes itself from one end of the earth to the other. It
preaches to all nations and to all worlds; and this word of God
reveals to man all that is necessary for man to know of God.
> Do we want to contemplate his power? We see it in the immensity
of the Creation. Do we want to contemplate his wisdom? We see it
in the unchangeable order by which the incomprehensible whole is
governed! Do we want to contemplate his munificence? We see it in the
abundance with which he fills the earth. Do we want to contemplate his
mercy? We see it in his not withholding that abundance even from the
unthankful. In fine, do we want to know what God is? Search not the
book called the Scripture, which any human hand might make, but the
Scripture called the Creation.
> The only idea man can affix to the name of God is that of a
first cause, the cause of all things. And incomprehensible and
difficult as it is for a man to conceive what a first cause is, he
arrives at the belief of it from the tenfold greater difficulty of
disbelieving it. It is difficult beyond description to conceive that
space can have no end; but it is more difficult to conceive an end. It
is difficult beyond the power of man to conceive an eternal duration
of what we call time; but it is more impossible to conceive a time
when there shall be no time.
> In like manner of reasoning, everything we behold carries in
itself the internal evidence that it did not make itself Every man
is an evidence to himself that he did not make himself; neither
could his father make himself, nor his grandfather, nor any of his
race; neither could any tree, plant, or animal make itself; and it
is the conviction arising from this evidence that carries us on, as it
were, by necessity to the belief of a first cause eternally
existing, of a nature totally different to any material existence we
know of, and by the power of which all things exist; and this first
cause man calls God.
> ...
> That which is now called natural philosophy, embracing the whole
circle of science, of which astronomy occupies the chief place, is the
study of the works of God, and of the power and wisdom of God in his
works, and is the true theology.
Yes, it looks like for many mathematicians, the confusion between stable conceptual foundation and eternal objective reality is too seductive to not fall in the illusion of identity of things locally indistinguishables.
