Is math based on a pure assumption?

I had never really thought about this, Jack, but apparently it is. Somebody has managed to define this dependency of math on underlying assumptions, but you need to read it again and again!

'Any mathematical system depends on a set of assumptions, and there is no way of escaping them. All we can do is to minimise them, to get a reduced set of axioms and rules of proof. This reduced set cannot be dispensed with, only replaced by assumptions of at least the same strength. Thus we cannot establish the certainty of mathematics without assumptions, which therefore is conditional, not absolute certainty. Only from an assumed basis do the theorems of mathematics follow.'

http://www.ex.ac.uk/~PErnest/soccon.htm

Mathematics is based on a number of assumptions. These are so simple that everybody agrees they have to be true. Some of the assumptions are things like 'if two things are each equal to a third thing, then they are equal to each other' and 'it is possible to draw a line between two points'.

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

As far as "pure," the modern mathematics curricula would make pure mathematics assumptions about real numbers and the existence of real numbers and sets of ordered pairs without explicitly adding the applied mathematics assumptions that real numbers could be represented by finite or infinite decimals, and without explicitly adding the applied mathematics assumption to sanction the use of real numbers ordered pairs and even triplet of real numbers as coordinates for geometric or physically lines, planes and space. So the assumptions of the modern pure mathematics curricula was too limited to serve real-world applications of mathematics. For clarity and precision in the exposition of high school and college mathematics, to the set and and pure number assumptions of pure mathematics, we need to add a second class of assumption...

https://en.wikipedia.org/wiki/Math

https://byjus.com/maths/analytic-geometry