I suspect this has been debated almost since arithmetic and mathematics started to become formal disciplines. So a very long time ago!
The difficulty I suppose is that although "we" invented the names of numbers according to our languages (one, two, three; un, deux, trois; etc.), and "we" invented bases and number systems (1,2,3,4; I, II, III, IV; etc.); the relationships seem to occur naturally.
Given a bag of 5 (cinq, V) apples, eat one and 4 are left. Or quatre, or IV...
Similarly even with the abstract concepts of algebra. (a+b)(a+b) always = a^2 + 2ab + b^2; and you can prove it by substituting any simple, real numbers for a and b. In any language, too.
Or, wander round the house with a fabric tape-measure, examining anything cylindrical and sensibly measurable; and you will always find the objects' circumferences / diameters = pi, within physical measuring limits of accuracy and (if you repeat the tests with the same objects), precision.
Ditto with the 3:4:5 right-angled triangle: it works whatever unit of measure, and provided the multipliers are consistent, any lengths obeying those two ratios.
What "we" have invented though, are sets of units of measure. Although the ISO-concocted SI version of the post-Revolutionary, French, Metric System is now the world's de facto, and usually only legal, system for virtually all technical and commercial work, the "sizes" of its fundamental units are still arbitrary inventions, save only for the system itself being arithmetically coherent - at some cost of practicality.
The point there though is that although the units are arbitrary however they were derived and defined, the mathematical and arithmetical relationships of real objects or processes they are describing do not change. That is so whether the real things are natural or artificial. SI merely simplifies the sums, though not simplifying Dimensional Analysis because we cannot divide one scientist by another!*
So, Pi is still Pi whether we use inches, metres or furlongs. Materials still change state at set temperatures whether we count degrees Kelvin, Celsius or Fahrenheit. The wave-motion equation Speed=Frequency.Wavelength is so, whether feet or metres per second. The numbers differ in each case but the relationships or mechanisms are the same.
' Mathematical techniques are a bit of a grey area in this regard. Did Newton and Leibnitz (independently) discover or invent Calculus? It did not exist as such until these two codified it, but the relationships it describes were always there in many natural mechanisms. Those gentlemen effectively discovered how to describe and quantify physical patterns - the numerical relationships - governing for example the acceleration due to gravity, or the link between the surface area and volume of a sphere. If they invented anything, it was how to describe the relationships; not the relationships themselves.
I suggest then that Mathematics as a way of describing natural laws, and hence its own internal nature, was discovered; but such things as Arithmetical Scales and Units, and their applications, were invented.
' ' '
*Dimensional Analysis.
Thereby hangs a tale of how some people can become utterly tied in knots by numbers (nots by knumbers?).
Some years ago, before the site first closed to posts then re-opened as a subscription-service, I was a regular user of the Q&A site "Answers-dot-com". Its Maths and Arithmetic section had a sizeable division devoted to converting between Imperial and Metric measurements.
I think most of those questions were from American school-children's homework. I would always give the method, but not the answer, as I did not want to encourage cheating. It is:
- Simply multiply the given value by the appropriate unit-conversion factor, easily found in many published sources.
Just one simple times-sum!
E.g. How many km in 40 miles? 40 X 8/5 = 64km. Approx! (The more accurate factor, 1.609, gives 64.36km. However, for most real-life, day-to-day trips, 5/8, which=1.625, is close enough; and that vulgar fraction is usually amenable to mental arithmetic.)
Sadly this area was plagued by two goons who delighted in making the exercise as hard as possible. They would introduce all manner of needless intermediate conversions, and try to invoke both Algebra (unnecessarily) and Dimensional Analysis (plain wrongly) - then make mistakes in their own, convoluted arithmetic!
