Professor Sherman Stein was a prominent mathematician and popularizer, and his book, Mathematics: The Man-Made Universe, is a modern classic. The subtitle “The Man-made Universe” already tells you that you’re looking at a clear exposition of “humans made math” in contrast to the “mathematics fundamentalism,” à la Professor Max Tegmark of MIT, whose tone seems to say mathematics allowed for reality and us.
This is of course a perfect “argument without end.” This is the kind of argument that should help a student to rethink their assumptions and not obsess about some once-and-for-all final understanding which can become an “idée fixe” (i.e., fixed idea in French, indicating being overly rigid or stuck).
In the preface to Professor Stein’s mathematics survey classic, he writes:
“We all find ourselves in a world we never made. Though we become used to the kitchen sink, we do not understand the atoms that compose it. The kitchen sink, like all the objects surrounding us, is a convenient abstraction.
Mathematics, on the other hand is completely the work of man.
Each theorem, each proof, is the product of the human mind. In mathematics all the cards can be put on the table.
In this sense, mathematics is concrete whereas the world is abstract.”(Sherman Stein, Mathematics The Man-Made Universe, Dover Publications, “Preface” Third Edition, page XIII, 1999)
Meta-intelligence tells you if views of what is real, what is concrete, what is abstract, what is man-made, what is mathematical, are so radically different depending on the interpreter or analyst, it makes prudent sense to keep various views in one’s mind and modify them or juggle them as you go along. Our ability as a species to nail down for eternity what the nature of mathematics, humans and kitchen sinks are and how they all interrelate, is elusive and tangled up in language, as Wittgenstein keeps saying.