Does 2 Plus 2 Always Equal 4? An Answer to Skeptics

Yesterday, I was at a prayer group, and one of my friends, Carlos, and I were talking almost incredulously about the sort of debates which went on in the early to middle ages of the Church. One of the major debates over Mary was this: was she conceived without sin (the majority position in both the West and especially the East), or was she conceived with sin, but then purified from sin at the point of ensoulment or quickening (this was the Thomist position, built upon the flawed medical/ethical belief that the kicking of the baby in the womb is a result of the soul entering the body). We were amazed because both parties were in agreement on about 99.9% of all things religious (whether or not they practiced these beliefs being a different story). There was no real question that Mary was born without stain of original sin, no real question that she remained sinless during her entire life, etc. In a thoroughly Catholic world, these things were taken for granted.

There are downsides to these sort of axiomatic foundations to debates. The first (major) danger is that one of the foundational beliefs is wrong. In the realm of science (unprotected by the Holy Spirit from error), we saw many drastically and fundamentally incorrect assumptions (the Earth is the center of the universe, e.g.) which were painfully excised from the community. The second danger is that, right or wrong, these unquestioned axioms are relatively defenseless. When all of the debate centers around minutae like how many days (zero or forty) after conception Mary was cleansed for life of sin, and all of the arguments are built upon the Scriptures and Tradition (interpreted by the Church), those arguing may not be in the best position to then answer the question, “And why should we beleive the Magisterium of the Church?” Or later, “And why should we believe Tradition/tradition?” Or later, “And why should we believe Scripture?” The third danger, related to this second one, is that human beings being what they are, this lack of defense will be made up for with force or coercive pressure of some kind. There are times when this is called for, of course. When someone refuses to accept the axiom, “murder is wrong,” or even, “murder is socially unacceptable,” it is neccesary to have a law in place to coerce them, and it is necessary at times to enforce said law. But these second and third risks, interconnected, can make the person relying upon axioms seem to be dogmatic and unthinking, when generally the opposite is true.

When I say the opposite is true, I mean that when you can agree upon certain axiomatic facts (2 plus 2 = 4, x=x, morality exists, moral good creates an objectively applicable ought, Christ is the Son of God and God Himself, we want to achieve y goal, etc.), you can move on to higher-level thinking. I was reading an interesting paper on how to explain numbers to robots. “For example,” said the paper, “if an agent has just determined that 2 is the greatest common divisor of 8 and 6, a highly idealized explanation dialogue might look like the following:”

Q1: Why is 2 the greatest common divisor of 8 and 6?
A1: 2 is the greatest of the common divisors of 8 and 6.
Q2: Why is 2 a common divisor of 8 and 6?
A2: 2 is a divisor of 8 and 2 is a divisor of 6.
Q3: Why is 2 a divisor of 6?
A3: There is a number that, when multiplied by 2, gives 6, and that number is 3.
Q4: Why is 2 times 3 = 6?
A4: Multiplication is repeated addition; 2 plus 2 is 4 and 4 plus 2 is 6
Q5: Why is 2 plus 2 = 4?
A5: When I count from 2 for two numbers I end up at 4
Q6: How do you know that
you will end up at 4?
A6: I counted two groups of oranges, with 2 oranges in
each, ending up with 4 total oranges.
Q7: What is 2?
A7: It is a number and the greatest common divisor of 8 and 6.
Q8: What is a number?
A8: Some examples are 2,4,6 and 8 . . . It is something that can be counted, added, multiplied . . . and something that can be the result of finding a greatest common divisor.

The answers are regressive to the most fundamental question: what is a number? And this is truly axiomatic. The definitional answer is circular: it’s a description that consists of providing examples and expressing its functions, functions which had already been demonstrated in previous answers. An over-simplified version would be this:

Q1: What is Tom?
A1: Tom is x.
Q2: What is x?
A2: What Tom is.

Circular. But perhaps necessarily so. There comes a point where you either accept the basic and irreducible premises (numbers exist), or you don’t. If you do, you can move on. If you don’t, you’re left always trying to reinvent the wheel. This is a radical oversimplification of what I think C.S. Lewis is driving at in his (boring but insightful) book Abolition of Man. If we don’t accept certain principles a priori, we can’t move on at all.

For example, imagine debating a person who refused to accept that mathematics were always true. They’ll grant you that 2 plus 2 equals 4 right now, but how do we know that 2 plus 2 always equals 4? It’s a non-testable hypothesis, what Peter Woit calls “not even wrong.” Yet if we don’t just assume (based upon what little bits we can observe, such as that 2+2=4 right now, and the last hundred times we checked) that 2 and 2 is 4, we cannot move on to any higher level mathematics. The Jabberwockian math skeptic may think you’re a fool for just assuming 2 and 2 is 4, but it certainly has the distinct advantage of letting you move on to calculus, while he’s trying to disprove addition.

In religion and morality, we’re increasingly running into these Jabberwockians: people who believe that all morality is subjective and individualized. Fashioning themselves skeptics, they fail to present a contrary theory, simply saying,”Why?” an infinite number of times to the objective norms which Christianity (or any system) offers. Generally, two major faulty assumptions are made: (1) that all individuals act upon, or wish to act upon, their desires, and (2) that might makes right. In theory, this system would offer little defense against Nazism or any other abomination. In practice, the individuals often unironically claim that they are still ” moral atheists,” or some variation thereupon. In other words, they still keep those parts of the Christian moral/ethical system which they can see a subjective value in keeping, then laud themselves for not violating it too egregiously. Typically, this means keeping the ethical system (do unto others), while abolishing the moral system (love the Lord your God) as archaic. Yet the ethical system, without the mooring of the moral system, is intensely dangerous. It usually devolves into crass utilitarianism: whatever benefits the most people is best. This sounds nice, but that’s the same sort of logic the Aztecs used to justify daily human sacrifices (sure, it sucks for the person whose heart is ripped beating out of their chest, but if the alternative is to risk even slightly the sun not coming up tomorrow, it’s worth it, right?), or that the high priest used to kill Christ (John 11:49-50).

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