My answer to: (For example if a higher being, hypothetically, were to inform the human race that 1+1 was not actually 2, and that maths is in fact an impossible and silly human invention, easily replaced by something else.) (To vote and comment on Quora, visit the link here.)
This is a pretty long answer, wherein I have tried to prove how each branch of mathematics does not need to be fundamentally right or wrong, but it just, is—for all to see.
Arithmetic. Take a chicken. Put another similar chicken with it. The new number of chickens is defined as “two chickens”. Add another. The new total number of chickens is defined as “three chickens”. Symbolically, 1 + 1 = 2, and 2 + 1 = 3. Nowhere is the scope of being fundamentally wrong. This is counting, and I put being able to count as the “sixth sense” we have just as the ability to see, hear, smell and touch. The thing that we see as a tree is defined a tree; the number of trees we can see is defined as one, two, etc. What I am saying is, arithmetic is physically verifiable, hence it doesn’t need to be right or wrong. Like, when we see something, we see something, it does not have to be bright or dark, red or blue, etc. Subtraction works the same as addition. Multiplication and division are defined as repeated addition and repeated subtraction, and since they are derived from physically verifiable operations by virtue of definition, hence are neither assumptions nor wrong. They just, are.
Higher arithmetic (say, fractions and decimals): Fractions as extended counting: If you are given an apple and a half of another apple, you can physically verify using your sense organs, that there are two apples, rather “two physically observable entities”. Now, just like your eyes can see but they can also figure out wavelengths of the light (i.e. colour of what they see), our counting sense can also, in addition to seeing there are two pieces, figure out that the second piece is not as big as the first, and hence shouldn’t qualify as one apple. Similarly, if you take a rectangle and divide it into six smaller boxes, and then paint two of the smaller boxes blue, we define the size of the blue portion as two-sixths of the bigger rectangle. A “sixth” defined as one of the six equal portions is same as defining a “tree” as a tree! What’s right or wrong about it! Hence, fractions are also physically verifiable, and hence nowhere fundamentally right or wrong. They just, are.
From here on, the following are not applications of arithmetic—these are roleplays of the numbers defined by arithmetic, for different purposes like to measure.
Geometry: Just like we can verify numbers in nature, we can also see geometric shapes. The ability to compare sizes is as fundamental as the ability to count and the ability to hear. (Can’t you tell which man is taller when one stands beside another, or which angle is bigger than another? You need a proof for that?) Now, out of the ability of comparing sizes originates the need to measure exactly, and hence geometry comes in as a logical product of physically verifiable measurements. However, here comes our first assumption: the use of numbers for measurements—the same numbers used for counting, is baseless. It is just a coincidence that we use the same counting numbers for measurements. Geometry, hence, gives a new identity to the counting numbers that were actually physically verifiable. Like we assign the name “Barack” to Barack, we assign the measurement “two feet” to what measures two feet. Geometry is a parasite that uses the language taught by arithmetic, and hence, adds meaning to the counting numbers. Like equations are a parasite that use the alphabet taught by English to represent variables. Again, geometry merely uses those numerals on loan from arithmetic so that it can compare sizes. The idea still remains the measurement of big and small, which is physically verifiable. By converting lines into number lines, we are making arithmetic the slave of geometry. But since geometry is verifiable and fundamental, we again come down to “neither right, nor wrong”; geometry just, is, and to express itself, it uses the language of numbers.
Note that geometry could also have existed if we’d used the alphabet instead of the number system. We chose the number system for convenience, because we already know how to add and subtract using them, and we found that those operations work in geometry too.
Irrational Decimals: We defined square roots, and imagined a number called square root of 2. Was it physically verifiable? We used geometry and discovered yes it was. Geometry gave an idea that numbers run continuously on a number line instead of discrete numbers we can choose randomly. There is a difference between “counting numbers” and “integers” or “real numbers”: the counting number “1” is an individual of arithmetic; the integer “1” is a slave of geometry who works on the number line.
Go on with every other branch of mathematics, and you’ll find that each either (1) is fundamental in itself (such as arithmetic), or (2) is fundamental in itself but uses another as its language, and it does so by redefining the characters of that “another” for its own purpose (like geometry uses the most fundamental counting system to compare sizes by redefining counting numbers as points on the number line), and hence no need of a proof,
(2) or, is defined in terms of another by virtue of definition (like calculus can be defined in terms of geometry), and hence no need of proof if that “another” was fundamental or proved.