Here we read:
. . . aren't all numbers inventions? It is not like they grow on
trees! They live in our heads. We made them all up.
The author of the quotation is introducing a discussion of the imaginary number i = the square root of -1. His point is that we are free to introduce this number since all numbers are inventions. So we can make up any number we like. The actual argument given is self-contradictory: The point of saying that numbers do not grow on trees is that they do not occur in nature. But if they live in our heads, then they are part of nature, because our heads ae in nature and what is in our heads is part of nature.
But let's be charitable. The argument the author is trying to give is something like this:
1. Numbers are not physical objects
2. Numbers are mental constructions.
That this is a non sequitur should be obvious. For there is a third possibility: numbers are abstract or ideal or Platonic objects. This third possibility is of course the actual view of numerous distinguished thinkers and is seen to be plausible once one considers the difficulties with the view that numbers are mental constructions.
Note first that an abstract object is not one produced by a mental act of abstraction. For present purposes we can say that an abstract object is any entity that necessarily exists but is causally inert.
Note second that a number is not the same as a numeral. One and the same number can be represented by different numerals. Thus the same number is denoted by the Arabic '9' and the Roman 'IX.' Numerals are signs of numbers, while numbers are not. So no number is a numeral. Numerals are typically physical (marks on paper, for instance); no number is physical. Ergo, etv.
We also note that 9 in a base-10 or decimal system is equivalent to 1001 in a base-2 or binary system. When I speak of the number 9 I am referring to the denotatum of the numeral '9' as this numeral functions within our ordinary base-10 system. That denotatum is the same as the denotatum of '1001' as the latter functions within a base-2 system.
One and the same proposition can be expressed by different indicative sentences. Thus the binary sentence '1 + 1 = 10' expresses the same true proposition as is expressed by the decimal sentence '1 + 1 = 2.' But if the two sentences are both interpreted relative to the decimal system, then they express different propositions, one true and the other false.
Our question is whether numbers themselves are mental constructions, not whether numerals are mental constructions. This is connected with the question of whether mathematics is in any sense conventional. No doubt notation systems are conventional, i.e. decided upon by human beings (or whatever other intelligent critters there might be elsewhere); but it doesn't follow that numbers or other mathematical objects are.
If numbers themselves are mental constructions, then they depend on our existence for their existence. Their existence is a mental existnce in or before our minds, and thus a dependent mode of existence. (Forget about extraterrestrial intelligences for the nonce.) The same goes for the truths in which they are involved. (Thus 7 and 9 and 16 are involved in the truth expressed by '7 + 9 = 16'.) But we didn't always exist. So if numbers depend ion us, they they didn't always exist. Consider a time before any minds existed, some time after the Big Bang and before the emergence of life on earth, say.
During that interval, the speed of light and the speed of sound were the same as they are now, and during that time the former was greater than the latter, as is the case now. Let 'c' denote the speed of light in a vacuum. C is identical to some number, which number depending on the units of measurement one employs. So c = 186,000 miles/sec (approximately). In the metric system, c = 300,000 km/sec (approximately). The point is that once the system of measurement is fixed -- which of course is conventional -- then some definite number is the SOL. Similarly with the speed of sound, SOS. Now
1. SOL > SOS
is true now and was true at the time when no humans existed. Of course, at that time the concept or notion or idea greater than (taken in its mathematical sense) did not exist since concepts (notions, ideas) cannot exist except 'in' a mind. ('In' here not to be taken spatially.) But the mathematical relation picked out by '>' existed.
For if it did not, then (1) could not have been true at the time in question. And the same goes for the relational fact of SOL's being greater than SOS. That fact obtained at the time when no minds existed. So its constituents (the numbers and the greater than relation) had to exist at that time as well.
Therefore, mathematical objects cannot be our mental creations.