Mathematics, objective truth and why 2+2=4 and not 2+2=5

There's a discussion on Twitter about the result of the arithmetic operation 2+2. It all started with James Lindsay who made a snarky remark to illustrate the point that there's The Objective Truth which is not a societal construct:

2+2=4: A perspective in white, Western mathematics that marginalizes other possible values.

Twitter users don't seem to be a fan of James Lindsay and wanted to prove him wrong by finding a way to "prove" that 2+2=5 and that everything is a societal construct. You can read Lindsay summary of the issue here, but first stay with me for a while. Full disclosure: I did not know who James Lindsay was until this whole Twitter storm.

This Twitter discussion, when you remove the facade of sarcasm, anger and spite, goes to the very core of mathematics - something I'm familiar with and very passionate about. So let me take you on a journey through crumbling foundations of mathematics and how they were rebuilt (or rather plastered together so they don't look like they are crumbling). I promise I will only talk science and I won't talk about politics and society until the very last section.

Do numbers exist?

When I was studying mathematics I had a class which started with the professor asking me this:

- Do you believe that real numbers exist?

I thought about it for a while and answered no.

- Good, this means that you're a sensible person. Do you believe that rational numbers exist?
- Not really, no.
- Great. Do integers exist?
- No...
- Do natural numbers exist?
- Oh, yes, for sure.

- ... and I thought you were a sensible person. Maybe you'll come round one day.

The professor knew very well that I love discussing foundations of mathematics and I knew that he liked to ask questions which sounded simple, but were in fact very complex.

This tongue-in-cheek discussion may seem odd on the surface, but it does sum up most of the 20th century advances in the foundations of mathematics and goes very deep into the understanding of the inner workings of science. Let's talk about that.

The barbers

The very first observation which shattered the foundations of mathematics was Russel's paradox. At the beginning of the 20th century it was generally assumed that you can construct a set out of every statement with one free variable, e.g. "set of all giraffes" (i.e. set of all x for which x is a giraffe) is a good definition of a set.

Russel's paradox is a set R that contains all the sets which are not members of themselves. Does the set R contain itself? If set R contained itself it wouldn't be a member of R. If the set R does not contain itself it's a member of R and thus contains itself. So the set R cannot exist. Hence, not every statement with one free variable defines a set and there have to be some rules on what defines a set and what doesn't.

This fairly simple explanation had a very profound implications for mathematics: if we cannot get the idea of a set right how can we talk about numbers and arithmetic and algebra and topology and... you get the picture. It has even wider implications for logic itself: not every statement can (and should) have a logical value. You can rephrase the Russel's paradox as:

A barber cuts hair of people only if they don't cut their own hair. Does this barber cut their own hair?

The axiomatic theory

This gave raise to the axiomatic set theory (and type theory, but I'll save that for another time). Even earlier than that Peano introduced a set of axioms that govern the number theory. So clearly the solution to the naive, intuitive approach to mathematics was to have a clearly defined, non-controversial set of statements which will be considered explicitly true (i.e. axioms) and rules by which we derive other true statements from these axioms (i.e. mathematical logic).

Peano proposed the following axioms ("successor" in this context mean simply "the next natural number" - although that has to actually be proven from the axioms):

1. 0 is a natural number.
2. Every natural number has a successor which is also a natural number.
3. 0 is not the successor of any natural number.
4. If the successor of x equals the successor of y, then x equals y.
5. The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

Some of the axioms are indeed non-controversial (2,3 and 4). However, the very first one proves to be somewhat controversial to this day. Even Peano, in the very first edition of his work, had an axiom that stated "1 is a natural number", which he later changed.

I did a poll to see whether people on Twitter think about the first axiom (should it be 0 or 1) and my followers seem to have very strong opinions on that subject.

In general it doesn't really matter for Peano axioms (apart from cosmetics) whether we will include 0 or not. The resulting model of natural numbers stays the same (of course, you have to exclude 0 in one of the resulting models to show that they are the same). 0 or 1 is just a token, a graphical sign for the first natural number. However, it does show that even the simplest axioms are not non-controversial and are not "objectively, intuitively true".

I didn't say anything about the axiom number 5 - in my opinion the most controversial of all of them. It refers to "a statement" and the truth of that statement. This leads us to the solution of the barber paradox and the meaning of truth in mathematics.

The orders of logic

The statement about the barbers doesn't have a logical value - it cannot be true and it cannot be false. One idea to deal with this is to consider this statement invalid, because it references itself - when you see the word "people" it applies to "barber", so the statement says something about it's own truth value.

In order for a statement to have a truth value it cannot reference itself. It can reference other statements and then we call this a second-order logic. For example a statement "0 is not the successor of any natural number" is a first-order logic statement. However, the axiom number 5 starts with "if a statement..." so it's a second-order logic statement, because it references to statements.

In order to break the circular reference a second-order logic statement may only reference the first-order statements and not the second-order statements. In the barber example it would mean that the reference to "people" didn't include the barber and then we can assign a logical value to it.

As you can see here what is considered a "statement with a logical value" gets pretty messy the closer you look.

The reason for axiom number 5 being controversial is that if we assume all of the Peano axioms we admit that even something as simple as arithmetic needs second-order logic. There are ways around that, but, again, that may be a good story for another time.

The Gödel incompleteness theorem

The Gödel incompleteness theorem stroke a final blow to the foundations already weakened by Russel and is my personal favourite theorem in all of mathematics. It states that no matter what axioms you accept as the foundation of natural numbers there will always be a statement which you won't be able to prove. Mathematician call this an incomplete system - it doesn't assign a truth value to all statements. Gödel says that every interesting system is incomplete.

The implication here is that you cannot create a perfect set of axioms - not only that the Peano ones aren't perfect, but no other ones will ever be perfect. Let that sink in for a while before we move to the second theorem - and yes, there is a second part.

While completeness means that all the statements have a truth value assigned to them, consistency means that every statement has at most one truth value - or, to put it in other words, there is no statement that is both true and false. Incomplete systems are not interesting to mathematicians - if you can prove that one statement is both false and true the whole logical reasoning collapses pretty quickly.

Second Gödel's theorem says that you cannot prove that a system is complete within that system. So you cannot prove that the Peano axioms don't lead you to a paradoxical statements within Peano axioms. You have to go outside that system. The search for a perfect, all encompassing system collapsed even more now.

OK, but this is just a theory(TM)

So your first reaction may be the same as the reaction of people who first heard about the Gödel theorems - the statements that this theorem talks about are probably not interesting from a mathematical perspective. The theorem may be true, but it's not practical in every day work.

Furthermore, if an unprovable statement comes up we can use our intuition and "objective truth" to figure out if it's true or not and add it to the axioms, should we need that statement. All is fine. I'm not a psychologist but I believe this reasoning is called denial.

Anyway, this status quo (again, denial) remained for a very short while. All was fine until the continuum hypothesis entered the stage. The continuum hypothesis is a fairly easy to explain statement. You may know that there's more real numbers than natural numbers. The number of real numbers is called continuum and the number of natural numbers is called aleph zero. The continuum hypothesis states that there's no set which has a size in between aleph zero and continuum.

Even simpler: there is no set which has more elements than there are natural numbers (or integers or rational numbers - they all have the same number of elements) and less elements than there are real numbers. Fairly simple. Also fairly unprovable (in the current axiomatic set theory). Also no human intuition as to whether it's "objectively true" or not. No analogue to real world that can give us any intuition.

So... that's it. You should feel defeated by now and question the existence of the "objective truth". If you don't please continue to the next section.

What about computer science?

Some of you may say: ok, but you know, continuum hypothesis is not something that changes my day-to-day life. If you're reading this there's a very high chance that you're a software developer or software developer-adjacent. If you are you may think that you're able to write a software which does anything you want and you shouldn't think about Gödel, continuum hypothesis and all that because it's just theoretical stuff.

My immediate response to it is: you're still in denial. Don't worry though, I also have a longer response.

Using a very similar method to Gödel's you can prove that you are not able to write a computer program which, for any given code in your favourite programming language, is able to decide if that program will stop or run indefinitely. To put it even more simply: you're (you, the human) not able to decide for every given code in any programming language if it will stop or run forever.

Now your reaction may be: wait, but if I just write printf surely I'm able to say it will stop. Yes, but the whole point is that you're not able to do it for every program. My favourite example of this is now spoiled thanks to Andrew Wiles, but let's assume that we all live in the 1980s for a moment and that the Fermat's Last Theorem is still unproved.

Fermat's Last Theorem is this: for every n>2 there are no positive integers a,b and c such that:

aⁿ + bⁿ = cⁿ

You can probably very easily write a computer program that will iterate over the values for a,b,c and n and check if the equation is true or not. Will this computer program ever stop? It will only stop if the Fermat's Last Theorem is false. Today we know that it will run indefinitely, but would you say the same thing in 1980s?

Of course, this is an imperfect example since Fermat's Last Theorem turned out to be provable (although there was a dispute on whether it may be unprovable), but it shows the complexity of deciding the halting problem even for very obvious and small computer programmes.

What about physics?

You may also be a physicists and say: this whole thing is interesting, but I'm a physicist so I'm not interested in either the continuum hypothesis or the halting problem or undecidable problems. I'm interested in figuring out how the universe works! I don't have a problem with a definition of "truth". I can use the non-fuzzy parts of mathematics.

My immediate response to it is: you're kind of right, but you're also in denial and by now you should know better. Don't worry though, I also have a longer response for you too!

Take a look at this actual theorem called the Banach-Tarski theorem:

Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

To put it in ever simpler terms: you can divide the ball in a couple of parts and then put them back together to make two identical balls. You can multiply balls. To infinity.

If you're a physicist you probably know that it cannot be done in the real world  - conservation of mass, along with many other laws of the universe, forbids it. However, it is true in mathematics. So how does this happen?

Well, the most straightforward explanation is that the three dimensional space used in mathematics is not the same as the one in the real world. It's not surprising, since the three dimensional mathematical space doesn't really have a concept of an atom or any other discrete particle.

So there is a mathematical theorem about the fundamental physical model - space - that is not really true in physics. At least not true in any practical sense and contrary to some of the physical laws. That would seem to suggest that there's a disconnect between "physical (i.e. practical, measurable) truth" and the "mathematical truth" even though physics uses mathematics for its research. I hope that gets you out of your denial.

What does it all have to do with objective truth and 2+2?

WARNING: this is where the more more opinionated part starts.

There seems to be no wide-reaching, Objective Truth in mathematics. There will never be one. There is truth within the constraints of a specific model. This model tries to correspond to reality as much as possible but it will never be intuitively or objectively true, because the very foundations - axioms - cannot be intuitively true and, what is even worse, even if they were, they wouldn't be able to prove everything. It will always escape the reality in some sense.

However, we all agreed on a set of rules and a set of facts from which we create the wonderful, messy, complex, sometimes frustrating world of numbers, sets, functions, arrows (I believe the alternative name for "arrows" is "category theory"?) and more. This is not bad, as long as we know that what we construct in mathematics is just that - our own constructions based on our own, flawed, imperfect intuition and perception.

To stir the whole objective truth mess a bit more mathematicians often play around with models that may be non-intuitive (e.g. non-Euclidean geometry) just so that they may see what interesting theories and properties arise. That brings us back to 2+2=4.

Summary

When it comes to a statement "0 is a natural number" there seems to be disagreement on how we want to model it and people are challenging either approach by presenting their opinions. Opinion is being the key word here - there is no objective truth to that statement, just arguments one way or the other.

When it comes to the continuum hypothesis it's even less clear - we don't have good idea how our brain wants to model the continuum hypothesis. That's ok though - we know that our models have to be incomplete and are based on how our brain perceives the world, which sometimes fails us.

This is, at least to me, the core of the issue - 2+2 is equal to 4 not because it's The Objective Truth, but because it's how our human brains want to model the world around us and because no one has a good reason to challenge it. If you ask me why we cannot say that 2+2=5 I ask you back "what additional interesting information would that provide?" If the answer is "none" let's stick to 2+2=4.

Just in case you were wondering, after all these years, I moved from my firm "yes, of course natural numbers exist" position to "maybe they do, maybe they don't, but it's fun to play with them". Maybe I'll come round one day.

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Side note to the whole blog post

What I talk about here is basically a whole century of mathematics condensed into one small blog post. I didn't touch on other solutions to many of the problems presented here: multi-value logic, type theory, non-Peano arithmetic, Church-Turing-Deutsch principle, Tarski's definition of truth and many, many other ideas that were created during that time. What I presented to you is a condensed version of the mainstream, which doesn't show you the nuance, but hopefully scratches the surface enough for you, dear reader, to look through that scratch and discover the weird, wonderful world beneath it.

Also, if you enjoyed this blog post and want to here more about the foundations of mathematics and computer science let me know on Twitter (@maldr0id).