
Stanisław Leśniewski: rethinking the philosophy of mathematics [pdf] - danielam
https://biblio.ugent.be/publication/4443772/file/4443780.pdf
======
Kednicma
It's always worth recalling _why_ we bother with set theory. Philosophical
objections like Leśneivski's are extremely valuable and good insights which we
cannot discard trivially. Maybe sets are not good things to study. The main
reason that we study sets today is because they are a place where we could
study ordinals and the rest of number theory. We know about two bananas, two
apples, two trees, etc. but what is two itself? Set theory provides a capable
if unsatisfying answer: Two is anything which is uniquely isomorphic to the
second ordinal number, which happens to be a particular finite set, and since
sets formally contain nothing but other sets, we can manipulate two as a set
without having to know about bananas, apples, trees, etc.

The modern way to talk about this stuff is via categorification; [0] is a good
high-level introduction.

[0]
[https://math.ucr.edu/home/baez/quantization_and_categorifica...](https://math.ucr.edu/home/baez/quantization_and_categorification.html)

~~~
spekcular
Set theory does not answer the question, "what is two itself?" This is a
common misconception. It just provides a way to translate statements about the
natural numbers into statements about sets, which then you can use for proofs
and formal constructions. It's an encoding, _not_ a definition. In particular,
the definition you give of two is not correct and I'd be interested to learn
who told you it.

Further, the "modern" way to discuss this is not categorification, despite
what Baez says; people who work and publish on the foundations of mathematics
almost universally do so in the language of set theory.

~~~
Kednicma
Since sets are 0-categories, we can't escape set theory when talking about
structures like the natural numbers. A natural numbers object is a feature of
a topos, preserved by topos functors (geometric morphisms). Nobody told me
this definition; it's something I had to absorb for myself when learning topos
theory.

Formally, let N be the natural numbers object (in some topos), let z : 1 -> N
be the zero arrow, and s : N -> N be the successor arrow. Then z;s;s : 1 -> N
is the arrow which chooses 2 as an element of the NNO. Since z and s are
unique up to unique isomorphism, so is 2. Moreover, since geometric morphisms
between topoi preserve finite limits, the NNO should also be preserved, and
that includes 2.

When the topos we choose/define is (equivalent to) Set, then we get the
standard ordinal-number definition of 2.

To use a pun, this lets us upgrade from Dedekind-categoricity to a more modern
and natural sort of categorical categoricity.

~~~
spekcular
Of course we can escape set theory (and categories) when talking about the
natural numbers. The concept of "two" predates the concept of a set by at
least a thousand years. People were happily manipulating and investigating the
natural numbers before set theory ever came along.

As I said previously, it is true that set theory provides a way to _encode_
the natural numbers as sets (or features of a topos, etc.), so that questions
about natural numbers can be stated as questions about sets. It is further
true that this endeavor can be incredibly fruitful, for instance for studying
the foundations of mathematics. But it does not mean that natural numbers
"are" sets (or objects in a topos), any more than Quicksort "is" a piece of
C++ code.

------
sanxiyn
See
[https://en.wikipedia.org/wiki/New_Foundations](https://en.wikipedia.org/wiki/New_Foundations)
for another alternative foundation of mathematics.

~~~
dr_dshiv
The original philosophical foundation of mathematics would be Pythagoreanism
[1].

For instance, that there is a special, transcendent meaning to "oneness" or
"twoness" — or more generally, that there is a basic harmony within
mathematics that manifests in the harmonies of the cosmos.

Here is a nice article on the sources of his mathematical contributions.
[https://www.sciencedirect.com/science/article/pii/0315086089...](https://www.sciencedirect.com/science/article/pii/0315086089900207/pdf?md5=f82cfebc9d937472cb2b6a4af44ae195&pid=1-s2.0-0315086089900207-main.pdf)

[1]
[https://en.m.wikipedia.org/wiki/Pythagoras](https://en.m.wikipedia.org/wiki/Pythagoras)

~~~
bawolff
Trying to find an axiomization of mathamatics and trying to find the
metaphysics of mathamatics seem like two very different projects to me.

~~~
dr_dshiv
Hmm, getting to the bottom of axioms feels a lot like metaphysics, but open to
why they'd be a different ballpark.

~~~
bawolff
In my mind, metaphysics is asking "why", axioms are asking "how"/"what"

~~~
dr_dshiv
That's fine. It's just that when you get to the very bottom of axioms, it can
get a little weird. Like, is the underlying basis the one, the nothing or the
all? Is it being or not being? That is metaphysical -- and has implications
for the foundation of any axiom, no?

~~~
bawolff
I'm not sure what it even means for the underlying basis to be one or nothing
or everything (basis of what specificly?). That doesn't sound like an axiom
thing to me.

