Because of the charged nature of this subject, I want to state up front that I think of myself as more conservative than Mark Levin and Rush Limbaugh. So, please don’t bail on this article yet, as I am not going to argue that 2 + 2 = 5, as to support an Orwellian view of reality (see the book 1984). I am not going to argue that 2 + 2 = 4 is a bendy idea; nor am I going to call it a lie. I have no liberal agenda (quite the opposite). To get to my conclusion, I know that some things I am about to write will make conservatives question my motives. But if you can hang on to the end, and suspend judgement till you get to the final paragraph, this article will truly delight your mind. The payoff is worth it, and by loosing one logical idea to roam about in your brain, you will experience a powerful chain reaction of well-reasoned thoughts. The payoff is worth it…
Despite what we teach kids in elementary school, there are many maths. But I am not saying this because I am an advocate of liberal education in America (far from it!).
Mathematics is many flowering and mushrooming inventions, expanding in many directions out of the creative minds of many humans. Each one is useful for its own problem domain. Yet, it would be a mistake to imagine these inventions as being handed down from heaven as divine revelation.
That humans invent math is itself offensive to religious people. So, in case you are religious, let me offer a mild proof. First, notice that people do invent things. Secondly, we don’t credit God with the invention of what we invent — for example, the car. This is not because we are atheists, but because God made man capable of inventing. Third, notice that men invent new objects used for all sorts of harm and wickedness that God would not invent (I will let you imagine such objects). In this article, when I speak of math as invention, the religious will immediately protest, but if they consider that God made humans as inventors, then I cannot be understood as being irreligious.
Mathematical systems are invented languages used to describe various human experiences. Again, the religious will protest. They may grant that humans can invent wicked objects. Religious people are even fine with humans inventing fake languages (like Klingon), but they never think of math as an invented language (and certainly not credited to men). But men do invent languages that work (again, I point to the silly example of Klingon, which is a modern made-up language that works as a closed system with its own invented grammar — i.e., its own logic).
There is no attack on piety in saying that mathematical systems are invented languages. Most of those languages work accurately and precisely within the closed systems for which they were intended. I am not saying these symbolic languages fail to truly describe what they describe, nor are their descriptions dubious. Rather, I am saying that each one works within the closed system where it is meant to be used. However, there is a fallacy afoot (the fallacy called equivocation) which extracts the propositions of certain of those closed contextual systems, and presses them as proof or samples of the largest context of all: the absolute.
I have in mind the statement, “2 + 2 = 4″, and how it is alternatively used as a proof of ultimate reality or as proof of divine reality. I believe in the ultimate and final reality of God (the only creator, the Trinity), but not because 2 + 2 = 4. So I am not denying the existence of God, nor arguing for relative truth. Not at all. My interest is that the statement, “2 + 2 = 4″ is often used as the irrefutable example of a universal absolute.
Consider a pretend conversation between a theist (someone who believes in a divine creator), and a moral relativist:
The theist: “God exists.”
The moral relativist: “How do you know.”
The theist: “2 + 2 = 4.”
Okay, the discussions are never that silly, but very often, 2 + 2 = 4 shows up in the proofs and appeals of the theist (I am not trying to mock any religious people, I just want to get to the point). The religious types may boldly and loudly assert that 2 + 2 = 4, as if that assertion automatically belongs to the set of absolutes to which their deity also belongs, or as a statement their deity delivered inscribed on stone (something they would never say about Klingon). Here, I want to examine that religious claim (no matter if the claim is explicitly or implicitly made).
Again, I am not denying that there is a God — may it never be! Rather, we must scrutinize the appeal to “2 + 2 = 4″, as it is often stated as some kind of irrefutable triumph of human thought and inquiry that has application in all realms of discourse (used as an axiom in theological proofs). My humble desire is to suggest that even “2 + 2 = 4″ has a limited context and application (even though those limits are quite broad), and is not to be ranked higher than experience, reason and context allow.
If anyone claims that “2 + 2 = 4″ has the force of an axiom useful for theological proof, they must establish what they mean by “2″ and “+” and “=” and “4″. These are words. These are words that may or may not have the same meaning when switching domains of discourse. Words have context. I am not denying that words work (they do!). I am not denying that words have real fixed meanings (they do!). I am saying that the meaning is context driven as much as it is dictionary driven. Both a context and a dictionary are needed (consider, for example, how the word, ‘draft’ has many contextual meanings as stated in a dictionary).
It is legitimate to say that in any conversation, we must establish what is meant by “2″ and “plus” and “equals” and “4″. We can’t move from a number system context to a theological one without naming the context switch (unless we want to be guilty of term switching — i.e., equivocating). So, let me circle back and restate the thesis in case I am less than plain:
There is a nearly universal, unquestioned and unfounded acceptance that 2 + 2 = 4 belongs to a super-context. It is a statement taken as one which defies any challenge from any lesser contexts. It is taken as the Super-Truth of all facts irrespective of any context. Since we were wee little ones, our parents have told us that 1 + 1 = 2, and 2 + 2 = 4. These rank among the cherished anchors of our childhood, reinforced in our education, and confirmed by our experience. We are free, however, to logically (and experientially!) disagree. Indeed, I wish to show that “2 + 2 = 4″ is not logically appropriate to all experiences and contexts.
I have done some work in the field of mathematics, and it is in that context that I developed a few thoughts about the matter.
2 + 2 = 4 is a powerful and useful philosophical construct that is not rooted in absolute truth, nor does it cross all boundaries, and ultimately is a creative expression of human experience that is accurate to its own context (that is, in its right context, it is reliable and fixed).
Now let me try to back that up. I will start with an apparent given. Namely, let’s assume that 1 + 1 = 2 is always true in all contexts of discourse. If 1 + 1 = 2, then by the law of non-contradiction 1 + 1 = 1 is false. However, I propose that 1 + 1 = 1 is true in this context:
1 + 1 = 1 inasmuch as 1 Fox + 1 Chicken = 1 satisfied Fox
It does not take years of mathematical study to figure this out. Even if we add things of the same type, 1 Rabbit + 1 Rabbit, we may get 5 Rabbits under the experience of multiplication (which is not the same as multiplication on the natural numbers, where 1 * 1 = 1, as in nature the idea of multiplication of the species from a pair is more like addition). In the case that 1 Rabbit + 1 Rabbit = 5 Rabbits, “+” is a term which may not mean what we think it means if we don’t account for the context. Addition is a verb or adverb, it is not a reality of its own. Likewise, numbers do not exist in reality, but are conceptual terms that have meaning according to the topic. I am not the first to observe this. Someone speaking of love and life has poetically said:
“I used to think the sum of one and one was two”, There Must be Something More, Theme song from Charlotte’s Web.
In all of this, we discover that “1″ is an adjective to describe something. “1″ is what it is, depending on what we are talking about. It is 1 Fox or 1 Chicken, and the + means adding the two together according to their kind. This way of describing reality is mathematical (within the closed system wherein the math is applied), but it is not the same math as what we find in the rulebook used in Algebra I courses (which is its own system).
And herein we discover that there is not one math, but many maths. For example, there are many “geometries.” There is Euclidean Geometry and non-Euclidean Geometry. I did not make this up, it is the way of things in the college curriculum. This is not an argument for philosophical or moral relativism. I am not pitting the different systems of math against each other, Euclidean vs. non-Euclidean (as if I have found some way to say that 2 + 2 = 4 is never true nor false)! If you think that is my argument here, you are changing subjects in your own head. I am merely arguing for the concept of Context, and against the fallacy Equivocation. Nothing I am saying here supports ethical relativism unless my words are wrenched out of context.
I can argue as I have (in part) because numbers are not abstract realities that have independent and real existence. You cannot go and get “5″ for me. “5″ is not a reality in and of itself; it is a description, and numbers on paper are descriptions of reality. They are adjectives. They are not real-nouns. I thus reject Plato’s Ideals, as there is not a thing called “5″ to which all sets of “5″ conform to.
An analogy may help here. Think of a musician and all those funny markings on paper they read as they play the piano. The music notation on the sheet is not the music, but the sound coming from the piano is the music. Likewise, math is not simply what happens on paper as we manipulate symbols — not any more than we would confuse sound with the dots and lines on the piano player’s ruled paper.
“2 + 2 = 4″ has more to do with our experience of a certain event in a certain context than it has to do with some ideal about an absolute statement that applies to all subject areas. 2 + 2 = 4 is a way we can describe certain ideas within a certain system (and in the right system, the integers, it is invariable and true).
Any particular mathematical system is not representative of the final absolute truth of all other domains of discourse, and is thus contingent. 2 + 2 is 4 on the integers. In fact (if we wanted) we could invent another system of integer rules where we don’t allow this (for more on this, I direct the interested reader to the study of Abstract Algebra and ring theory, and more tangentially, to modulus arithmetic).
I am not trying to pull a fast-one over you. I am saying that when you were a kid in grade school, we were speaking like children about one subset of language (arithmetic). But we are not children anymore, so we can expand our understanding of words to see that they have meaning according to their context. Words work. And they work accurately and precisely as to their context. I am not arguing against 2 + 2 = 4, and I don’t know of any logical system where 2 + 2 = 5 on the naturals, wholes, or integers (etc.).
I tell all of my math students that the rules of math are the inventions of people to describe systems, and we are free to make new rules — and in making new rules we may invent a clever and useful system of thought. Mathematicians are constantly inventing new ideas and rules. Math is not a set science because there are many domains of discourse and many humans discoursing. The high-school idea of math needs to give way to this more robust grown-up view of maths as being languages.
Mathematicians often make up new ways of thinking (to solve problems or to invent new thoughts). This kind of creativity happens all the time. For example, there was Newtonian Physics and then there came Einsteinian Physics. The best we can do is say that these “physics” conform to our current or observed experiences, and another Physics may be invented to describe reality. Newton is not eternally right across all domains of discourse, nor were his laws. There was a time when energy (which he said can’t be created or destroyed) was created — if we deny this, then we are forced to postulate that energy is eternal (whatever that would mean). Another way to say this may be less controversial: We have no sure facts to prove that Newton’s laws have always worked, and now Einstein tells us that things are different anyway.
Take another example of “infinity” as it is used in Algebra I or in math books. We talk about infinity as if it has a given meaning on the integer numbers (namely, counting on and on forever). But then Cantor came along and invented different kinds of infinities. It turns out that “infinity” is not the thing we always thought it was. There are countable and uncountable infinities now, thanks to Cantor’s invention. See, Math happens! There are maths, not math.
I am writing this, in part, to help take away any false assurance that math is some pure and absolute discovery that sits alongside divinity. I would like to extend my thesis to say that maths are invented, not discovered (for example, Russell gets credit for his set theory).
I like mathematical languages. But I don’t kid myself to imagine that they cross domains and take the place of divinity (contra those who claim that mathematics is the one source of pure absolutes). Mathematical language is not inherently false (I have not argued that), but neither is the language of the integers appropriate to all contexts.
If a person wants to dethrone God and install math in His place, they cannot claim that they have math as ultimate truth. Math does not rise to the level of deity (unless you want to invent a deity). To reject Jesus while claiming to be mathematically sure about math and science is to deify math. Secular mathematicians are known to do this. I encourage them, if they reject Christ, do not do it because you have found a universal truth in math. You have not. “2 + 2 = 4″