Let’s start with a quote:
Mathematics is a game played according to certain simple rules with meaningless marks on paper
Who do you think wrote this? An angry school kid, frustrated with algebra? An educational reformer? A person with a lifelong hatred of math?
No. This was written by a very influential mathematician named David Hilbert. Not only did he write this, he didn’t intend it as an attack on mathematics.
This article will explain why.
The key here is what he meant by “meaningless”. This is a problematic term, because it is often used in a demeaning way. People have “meaningless” jobs or “meaningless” lives. But one needn’t take it as an insult. So let’s explore the meaning of “meaning” together.
We’ll start with claim:
Cows are carnivores.
This statement is false. But how do we know it’s false? Well, because we know what a cow is and what carnivores are. We’ve seen them in all sorts of situations, and have all sorts of associations with them. So when we see a claim like this, we consult our memories of experiences with cows, or if worse came to worse, go out and observe some cows. In short, we conclude that the claim is false for semantic and empirical reasons.
Now let’s take another statement:
Blimfings are freemps.
Is this statement true? Well, I don’t know. Those are meaningless terms. What’s a Blimfing? What’s a freemp? Well, maybe if I had a definition, that would help. Ok, so let’s say I’m given the following definition:
Blimfings are gloomphs
Well, that was useless. Sure, I technically have a definition, but the terms it used were meaningless too, so the whole thing is still meaningless. For Blimfings to be a meaningful word, I must have some experience or associations with it, whether with the word directly or with its component defintions.
Well, what if I get another bit of info:
All gloomphs are freemps.
Does that help? Well, actually it does! To see why, let’s list the 3 things I’m told about Blimfings:
All gloomphs are freemps.
Blimfings are gloomphs.
Blimfings are freemps.
Well, I don’t need to know what a gloomph, freemp or Blimfing is. I know that if the first two lines are true, then Blimfings must be freemps. The first two lines are premises, but we can also think of them as hypotheticals or even axioms. The last line is the conclusion, and it follows from the axioms.
So the key here is the logical relationship among the terms. Now I can say that the above is true because it’s universal (that’s why I know this is true without even knowing what the terms mean). But it actually helps that the terms are meaningless, because if they had meaning, I might have just consulted my memory instead of looking at the structure of the argument. It’s precisely the meaninglessness of the terms that has forced me to consult the structure and recognize the universal pattern lurking within.
While the meanings of Blimfings, groomphs and freemps don’t matter, the shapes of their strings do. I see freemps repeated in key parts, and since the shape is the same, I can draw certain conclusions. At this point, I’m doing formal or symbolic analysis, because it’s not the meaning of the strings that matter, but the shapes, or forms.
And that’s what math is. It’s a formal, hypothetical argument. But don’t take my word for it. Let’s take another influential mathematician and philosopher – Bertrand Russel:
Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. […] We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics.
Since the terms don’t matter, we just need placeholders, so let’s use variables, which are about as meaningless as you get. We’ll use X, Y and Z instead of Blimfing, gloomph and freemp:
All X are Y.
Z are X.
Therefore, Z are Y.
This is the structure of a logical form called the syllogism, and it’s usually presented with X=men, Y=mortal and Z=Socrates:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
Let’s dig further into what symbol manipulation entails, to show how it can provide a very simple foundation for proofs, and how proofs can be conducted with no regard for the meanings of any of the terms. First, let’s imagine we have a simple string substitution system:
X → Y
Y → Z
Prove: Y = X
The first two lines simply say “if you encounter an X, replace it with a Y, and if you encounter a Y, replace it with a Z”. We call these our axioms, and you’ll see why shortly.
Now, given the above, how does the proof of Y = X unfold?

We start with Y = X and ask if we can substitute anything. Yes we can; one of our axioms is Y → Z, which means we can replace a Y with a Z. So we do this and we get…

Z = X. Well Z and X are clearly not the same symbol, so we ask if we can substitute anything else. The answer is yes, we have an axiom X → Y, so we perform that substitution and we get:

Z = Y. Again, this is not true as the symbols differ, so again we ask if we can substitute anything. Yes we can, we have the axiom Y → Z. We substitute that and we get…

Z = Z. That’s the same symbol on both sides of the equal sign. This means we are done and the derivation is true.
Now if we couldn’t perform any substitutions and different symbol strings were on either side of the equal sign, then the proof would be false.
That’s all there is to it. Substitute strings and check for identical strings on either side of the equal sign. This sounds simple, but it’s very powerful. For instance, let’s use this system on our syllogism:
All men are mortal
Socrates is a man
Therefore, Socrates is mortal
Since the first two lines are axioms (or premises) they get coded as the rewrite rules (now you see why we call those rewrite rules our axioms), and the conclusion is the thing to be proven:
man → mortal
Socrates → man
Prove: Socrates = mortal
Now, we’re substituting things that are equal, but also things that belong in the same group, so the substitution isn’t exactly for equality, but in this instance, that doesn’t matter. Let’s have a go. To keep the output condensed, I’m showing the rewrite step applied before each output using X → Y on an indented line:
Socrates = mortal
Socrates → man
man = mortal
man → mortal
mortal = mortal
As you can see, this simple substitution system – that’s ignorant of things like mortality, humanity and famous philosophers made short work of this syllogism.
Ok, so playing with syllogisms is one thing, but can we do more? Can we prove things about math?
Yes we can!
Let’s try a trivial example, proving basic addition equivalences like 2+2 = 4. We need our system to perform replacements so things like 2+2 and 3+1 end up with identical symbols for 4. But we also don’t want to cheat by hardcoding, especially since we want to be able to prove any simple addition equivalence.
After some thought, we may realize that any whole positive number > 1 is just a sum of 1’s (this is basically Peano’s axioms). So with that in mind, what about this?
2 → 1+1
3 → 2+1
4 → 3+1
Prove: 2+2 = 4
Now I could have directly coded 3 as 1+1+1, but thought it would be more amusing and less typing to do it this way. It still comes out to the right thing and it doesn’t know anything about numbers, and you’ll see how once we run the proof.
So with that in mind, let her rip!
2+2 = 4
2 → 1+1
1+1+2 = 4
4 → 3+1
1+1+2 = 3+1
2 → 1+1
1+1+1+1 = 3+1
3 → 2+1
1+1+1+1 = 2+1+1
2 → 1+1
1+1+1+1 = 1+1+1+1
Did you see how it replaced 4 with 3+1, and then later replaced the 3 with its definition of 2+1, then later replaced the 2 with its definition of 1+1, which ended up making 4 1+1+1+1 anyway? It knows nothing of addition, but our rewrite rules take care of that. And in fact, since 1+1+1+1 is the identical string on both sides of the equal sign, this is true and it’s proven.
We could also use the same axioms to prove 3+1 = 4, 2+1 = 3 and 1+1 = 2 to name a few. We can even prove commutative examples like 3+1 = 1+3. This is one reason we didn’t hardcode to get the proof, but chose a representation in which the shapes of the symbols showed equivalences.
Now these are all trivial examples, as I’m trying to demonstrate the concepts, but for those who are interested in meatier examples, I recommend checking out the Mizar system, which uses substitution in its proofs and has formalized a large body of mathematics. In fact, you can download the system and play around with it yourself, and even check out some proofs.
Hopefully, this article and sample system shows you not only how mathematics can be regarded as a meaningless game, but how this isn’t an insult. Meaninglessness here simply means the symbols have no meanings, but are placeholders, and this is what enables the flexibility of formal or symbolic manipulation.
Calling mathematics meaningless therefore, is not an insult, but a term of endearment.
I’m a mathematical empiricist (or perhaps more accurately, a semiempiricist). I think the “unreasonable effectiveness” of mathematics tells us that its foundations are based on relationships that exist in the world.
Granted, many abstract mathematical structures have no real world correlates, but that’s also true of scientific theories. We just judge mathematical structures with different values. Scientific theories are judged by how well they predict observations while mathematical concepts are judged by their coherence.
Yet, it’s not unusual for abstract math to unexpectedly become useful in scientific formulations. Of course, sometimes it goes the other way. From what I understand, string theory has done more for math than it has for actual physics.
Graham Priest, an expert in logic, thinks of logic as theories about fundamental relationships. I think the same thing could be said for mathematics.
I agree on all counts. I think math and logic arose as applied, practical tools, and “pure” mathematics was a further abstraction. With that said, I think it’s accurate to look at math as an abstract system, as that doesn’t imply anything about the history of math or how it arose.
But we may still be able to apply the notions of abstraction or symbol manipulation with concrete math. Even at its most concrete, the notion of a number is an abstraction (2 sheep, 2 cows, 2 apples, abstract them away and you get 2). Once that’s there, then 2 can be seen as a series of marks as one sort of canonical representation, from which those series of marks can be manipulated in purely formal terms, without any reference to meaning or the real world. That’s what I hoped the proof of 2+2 would show.
Also, with regards to pure math finding application in the real world, I do find that fascinating, but is it that strange? Even pure math is concerned with relationships; is it so surprising that occasionally a relationship is found that maps onto a realworld one?
Ultimately, it is disingenuous when pure mathematicians scoff at applied math.
In my article, I didn’t mean to imply that math originally arose as an abstract study. I was just interested in how we could define math today, and how meaning and abstraction relate.
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