What is the fundamental "boiled-down" element of math?

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Apple30

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For example, everything in programming can be boiled down to simple logic-gates (NAND) which perform the most basic binary manipulation.

I wonder, what is the fundamental "essence" of math? Is it arithmetic?

I've read one person online saying "Everything boils down to addition", another says "numbers", another says "sets and maps of sets" (whatever that means), another says "addition, subtraction, multiplication, division, and modular forms".

Interested in hearing peoples thoughts on this.
 
IMHO, math just does not boil down. But some might mention mathematical logic or set theory.The way I see it, one comes up with a set of axioms and then people keep "discovering" interesting theorems. If they discover enough of them they get an interesting math field, which, in turn, can help to build other fields. E.g., group theory and general topology lead to algebraic topology.
 
In my personal opinion the following seem quite fundamental (sorry that I haven't restricted myself to a single point)...
  • Implication. From a few "seed" axioms a huge tree of factual implications can often be discovered
  • Proof. Transformation of a conjecture into a theorem
  • Extension. An area (of maths) can often be extended in a back-compatible way via the addition of one (or several) new axiom(s)
  • Abstract thought. Much of mathematics is independent of our physical environment, our decimal system, our own language, etc. But it turns out to be very useful for making the most of the physical environment.
  • Clarity. Discovery of a notation that aids the advancement of a specific area.
  • Generalisation. Making a particular discovery useful in a wider variety of situations.
  • Mapping. Transformation of a system/ problem into another equivalent form
An interesting note: I notice from my children's textbooks that some things that I was taught in computing classes now seems to be taught within further maths classes. Specifically logic (boolean algebra) and algorithms (including sorting, tree navigation, graph - shortest path etc, complexity of algorithm P/ NP etc). I guess there's a strong crossover/ relationship here and mathematics certainly reaches into many other areas of study too.
 
I mean this conversation definitely brought Einstein's quote to mind “Pure mathematics is, in its way, the poetry of logical ideas.” -Albert Einstein
And to expand subjectively what i have found most commonly as an answer from my students when inquired about their keen interest in mathematics more than other subject- it feels like its the guarantee of control over the numbers, the functions and the guarantee that if the formula is correct, the answer would be same for everyone. I was also reading a report by EPW in india how mathematics is one of the most preferred subjects by the under-served- while the obvious reason is that mathematics can sometimes side step the hierarchical stuff inherent in languages etc, i think another reason might be one of these concrete control. So I am extremely humbled by the subject and would argue that one of the essential parts of mathematics is play- be able to manipulate numbers and the have some interesting and absurd qns- like one about cockroaches in an island- as well some about more fruits than anyone can have.
 
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