Spivak's Calculus Assumptions 2

Question

Spivak's Calculus is known for being the best Calculus book when it comes to rigour, but I think I've found an assumption he makes during his very first proof!

The axioms are:

P1 (Associativity of Addition): (a + b) + c = a + (b + c)

P2 (Additive Identity): a + 0 = a

P3 (Additive Inverse): a + (-a) = 0

P4 (Commutativity): a + b = b + a

The Proof:

(1) If a + x = a,

(2) Then (-a) + (a + x) = (-a) + a *(adding (-a) to both sides)

(3) Then ((-a) + a) + x = 0 *(using P1 and P3)

(4) Then 0 + x = 0 *(using P3)

(5) Then x = 0 *(using P4 and P2)

Conclusion: If a + x = a, Then x = 0

My problem is in the very first step from (1) to (2). Where is the justification for adding (-a) to both sides? Seems to me if we are going to do such formal proofs with such basic axioms for such simple theorems, such an assumption is quite severe? May as well assume the conclusion! To clarify, mathematically the assumption is:

If a = b, Then a + c = b + c

You can only add numbers to both sides of the equation if you establish this truth (either as a theorem or an axiom). I don't think it's possible to prove it with P1-P4.

So should it be included as one of Spivak's axioms?

Am I missing something?

Answer 1

You are right, Spivak hasn't defined what the symbol $=$ means and well a plethora of other problems if you want to be absolutely formal. But I believe you can make the exact same objection about any step of the proof. What does it mean mathematically for something to follow? Why if (2) then (3)? The thing is, Spivack is using another structure implicitly which is first order logic, that provides a way to structure mathematical thought unambiguously. In https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms you can see that you can define the symbol, but well in the same page you can see it still not as simple as just making a list of more and more axioms. You can see that it follows from substitution for formulas. I would still consider Spivack rigorous in the same way Euclid's Element is rigorous, the things that are implicitly assumed are for the purposes of the book "universal", even though by modern standards of formality they clearly aren't. Though you would need to read on logic and set theory to have the modern perspective and it is no simple task.

Answer 2

Let me suggest following explanation variant: just before these axioms Spivak defined operation addition "for numbers-the sum $a + b$ exists for any two given numbers $a$ and $b$"(p.3). Operation here means function $X \times X \to X$, so if we take one number, $a+x$ for example, that addition to it other number, for example $-a$, will give same result. So if $a + x = a$, then $(-a) + (a + x) = (-a) + a$ from addition unambiguity.

Answer 3

The wording is "adding $-a$ to both sides", but that is not what actually happens. You just use $a+x=a$. Then $$ (-a)+(a+x)=(-a)+a $$ simply because we are replacing $a+x$ with $a$. The "truth" that Spivak is using is that if two things are equal, you can replace one for the other, which is the whole point of equality.