Mathematics and Empiricism

Ape in the coat
Fri, 24 Apr 2026 16:02:00 GMT

In Does the Universe Speak a Language We Just Made Up? Lorenzo Elijah, PhD shares his fascination with math and echoes a common idea among philosophers that the “surprising efficiency of math” is a problem for empiricism and physicalism:

Mathematics is GOATED. You can use it to do your taxes or put a guy on the Moon.

But it’s also very weird.

If you’re a hard-headed empiricist or a flat-footed physicalist, the idea that a bunch of nerds can discover deep truths about reality from the comfort of their armchairs, should strike you as absurd.

And yet, mathematics is the backbone of all the hard sciences.

Personally, I’m a big fan of empirical observations and physicalism does appear to me as the most probable ontology. I’ve written several posts critiquing metaphysical reasoning for trying to guess nature of reality from the armchair. And yet, I don’t see anything weird in the fact that math is the “backbone all the hard sciences”.

This may have something to do with the conflict of interest. My university diploma claims that I’m a mathematician, after all. Alternatively, maybe I have a better model of math than Dr Elijah. Confusion is the property of the map. If reality seems weird to you, then it’s you who has a problem, not reality. I’ll let my readers to deside which one of these factors is more relevant here.

Reality and Fiction

In his post Dr Elijah argues against mathematical fictionalism. He describes it like this:

On this fictionalist view, mathematics is just a language like any other. What makes mathematics special is that it is far more precise and general than any other language we invented. Its utility in science is exhausted by the fact that it allows us to be extremely precise and logical when describing the natural world.

I would not call myself a fictionalist, however, I think the view described here is basically correct, capturing core truths about the nature of math. Indeed, mathematics is a very precise language that allows us to talk about specific things instead of something else by logically pinpointing them. And then we can preserve truth relations and reach conclusions that follow from the premises.

Yet, Lorenzo Elijah thinks this view is problematic as it denies that mathematical objects “really exist”.

To be clear, no one denies there is a conventional element to mathematics. We didn’t discover the plus sign floating in space. The symbols and notation we are familiar with was obviously something we made up. The dispute is whether the things those symbols refer to, like the number 4, really exist or not.

Fictionalists say they don’t.

I think we have decisive evidence to reject fictionalism about mathematics.

Before we engage with his argument, I’d like to share one tip for disentangling standard philosophical confusions. A lot of philosophical categories are a grab-all-bags of multiple meanings linked together for semantic reasons. And “real” is one of them. So, whenever philosophers talk about realness of something - pay close attention to what exactly is meant.

To Lorenzo Elijah’s credit, he tries to preemptively address this by explicitly stating that we are not talking about “plus sign floating in space”. This is commendable, but not enough. We haven’t tabooed the word “real”, haven’t got to the substance of the meaning that is being explored here.

What does it mean that a referent for symbol “4” really exists? What even is this referent?

1. If the question is whether there are at least four objects of some kind in our universe, then the answer is “Yes”, and I don’t think that an average fictionalist is going to challenge this.

2. If the question is whether the minds of people have a particular notion that is represented by symbol 4, which is systematically helpful for the purpose of reasoning about objects in our universe, then the answer is also “Yes”. Once again, I do not expect fictionalists to disagree here.

3. But if the question is whether this notion of “fourness” exists somehow independently from the physical objects of our universe or minds reasoning about them, in some separate realm of pure ideas, then fictionalist’s answer, as well as mine, is a clear “No”.

So, it seems that 3. is our crux of disagreement. Now let’s get to the argument that Lorenzo Elijah presents to resolve it.

The Argument

Here is Elijah’s argument against fictionalism:

From the armchair, mathematicians “discovered” a crazy weird object based on how beautiful and interesting this structure was. Years later, physicists independently discovered that this crazy structure just is the structure of quantum mechanics. From the armchair, mathematicians discovered something true about the fundamental nature of the physical world. If that doesn’t shock you, then maybe you need to read that again.

In case you still don’t see it, imagine how you would react if one of our space probes discovered a new planet with intelligent life. We go to visit, and we discover that the whole planet perfectly matches J.R.R. Tolkien’s Middle-Earth from The Lord of the Rings. Characters, history, everything. Our astronauts sit down with Aragorn and Gandalf and the two go on and on about four brave hobbits from back in the day.

I’d wager that you would find this miraculously surprising. What are the odds that Tolkien could fabricate a story that would perfectly match the story of a real alien planet? I don’t have precise numbers, but let’s just say I wouldn’t take those odds.

The situation with mathematics is comparable. If mathematicians are merely inventing elaborate theoretical constructs, independently of any practical concern, then it is unreasonable to expect that any of these constructions would perfectly match the underlying nature of reality. And yet they do—again and again.

This is why I find the idea of mathematical fictionalism implausible. If mathematicians are just inventing math in the same way that Tolkien invented Orcs, then we shouldn’t expect the world to mirror that math any more than we should expect to find an alien race that perfectly fits the bill for Tolkien’s Orcs.

If on the other hand, mathematics is discovered, then it is more likely that the natural laws of the world are “written in the language of mathematics.”

I agree that if mathematicians indeed were inventing/discovering math from their armchairs without any contact with reality and then it just so happened to perfectly coincide with the way reality is, that would be extremely surprising. Which is a huge hint that it’s not what is actually happening.

Remember, your strength as a rationalist is being confused by fiction more than by reality. If something is extremely unlikely to happen, then most likely it didn’t, and something is wrong with the theory that it did. In our case, with the argument above. So, what’s wrong with this argument?

Several things.

Math Is a Language, Not a Story.

First of all, let’s notice the core problem with the Middle-Earth-planet analogy. Math is a language that can be used to describe reality. Not a story about reality. This difference is quite crucial.

If we come up with some story and it just so happens to be true, it’s an incredible coincidence. If we come up with a language that can be used to describe what’s true - that’s just how languages work. Wouldn’t be a huge surprise if objects on the other planet can be described via the language of Tolkien’s elves, would it? Even less so with math which is much more general by design.

We can still be somewhat surprised if the language has necessary words to describe certain parts of reality before they were encountered. But this is an improbability of a much smaller degree, reduced by the fact that creating new words in this language is an interesting and rewarded activity, which people tend to do for its own sake, as exactly is the case with mathematics.

Math Is Not a Single Thing

When people are baffled by how well math describes the fundamental nature of reality, they tend to forget that math is a compilation of many different axiomatic systems. That it consists of multitude of “crazy weird objects”. Another failure of the Middle-Earth-analogy is that Lord of the Rings is the work of Tolkien, while the structure of quantum mechanics is described by yet another mathematical object, no more special than any other.

Imagine if Tolkien wrote thousands upon thousands of books, essentially brute forcing through all possible coherent settings. And then some planet with life happened to resemble one of them. That would be much less of a coincidence, wouldn’t it?

The same principle applies here. It would’ve been super impressive if mathematicians could a priori deduce which mathematical object in particular describes the fundamental nature of reality - to the point that we would indeed have to rethink the importance of empiricism as well as the established laws of thermodynamics.

On the other hand, when you try to explore all kind of premises and their conclusions, it’s no surprise that one non-specific of them happened to fit reality. We would still need to do the actual work and look through all of them, comparing them to the reality to figure out which one it is.

Realism Doesn’t Help

Last but not the least, is that, even if fictionalism performed poorly in accounting for this aparent coincidence, realism does no better.

Let’s suppose, for the sake of argument, that there exists a separate platonic realm about which mathematicians are reasoning. And then it just so happens that reasoning about this realm are applicable to our physical universe. Why would it be any less of a coincidence?

Previously we were surprised by the correspondence between our armchair reasoning and physical reality. Now we are, likewise, surprised by the correspondence between physical reality and platonic realm. The fact that our reasoning got an external referent doesn’t make this correspondence less improbable.

In fact, we would now also need to account for coincidence between platonic realm and our mathematical reasoning. How comes our brains have access to this separate realm, if they evolved in the physical reality? We can invent some just-so dualist explanation, but it will come with extra complexity penalty.

Imagine if someone tried to explain the similarity between Middle Earth and some discovered planet A by the fact that Tolkien was in contact with aliens from some completely different planet B. Not only this would not reduce the improbability of the initial coincidence, but it’s also an additional extraordinary claim that one has to justify.

What would help, is if there was some connection between planets A and B. If platonic realm was connected both to the mathematicians reasoning and to the physical world. But this connection is also an additional assumption which we’d need empirical evidence to back up. If a theory postulates something like this, it has to make some testable predictions about the nature of the connection, that have to come true to outweigh the complexity penalty.

And if we are ready to consider that there is a connection between our mathematical reasoning and physical universe through the platonic realm… why involve the platonic realm at all? Why not consider that there is a direct connection between mathematical reasoning and physical universe? That’s a strictly simpler theory, isn’t it?

Math As a Generalization of Observations

Wait… am I claiming that math is empirical? But isn’t it clearly absurd? Everyone knows that mathematical truths are certain while empiricism produces only probabilistic estimates!

I’ve briefly talked about this misconception in Give Skepticism a Try:

Thirdly, math is merely a truth preserving mechanism, a study of which conclusions follow from which premises for a certain definition of “follow”. It’s generalized conditional knowledge, not fundamentally free from the uncertainty but merely outsourcing it to the moment of applicability. As a result math can’t actually prove anything about real world with perfect confidence. A mathematical model may be making confident statements conditionally on all the axioms being satisfied, but whether or not the reality satisfies these axioms is an empirical question.

But let’s focus on this a little bit more. Consider these empirically testable hypotheses:

If there is one apple on the table and you put another apple on the table, there are now two apples on the table.

If there is one apple on the table and John puts another apple on the table, then there are two apples on the table.

If there is one orange on the couch and Mary puts another orange on the couch, then there are now two oranges on the couch.

If there is one apple on the table and an orange on the couch, and no other fruits on either of them, then, between a table and a couch, there are two fruits.

We can write a lot of such statements. For every type of object, for every type of actor, for every type of counting mechanism. Or we can generalize our observations and say:

For every object and process that work exactly like addition of natural numbers, one of such objects and another one of such objects submitted as an input of such process result in two of such objects.

Or even simpler:

1+1=2

This is a tautology, of course. True by definition. True only for things that it’s true for; false otherwise. This is where the absolute certainty comes from.

Valid theoretical reasoning conserves truth. It allows us to outsource improbability from one part of the theory to another, while keeping total the same. With math we simply outsource all the improbability to the question of whether a particular real-world scenario fits our mathematical model.

This truth preservation property can be very powerful. We can deduce consequences of some assumptions and immediately apply them to any part of the real world about which we managed to become quite confident that it satisfies these assumptions. The latter is something that still require us to go outside and look and isn’t something we can be totally certain about.

And I understand why it may feel magical. But this is a magic rooted in empiricism about our physical reality. Only empiricism can reduce the improbability of a particular part of reality fitting specific mathematical axioms. Moreover, our ability to reason about properties of the universe and generalize patterns is the result of evolving inside this universe that has such properties in the first place.

Therefore, there is nothing weird about mathematical efficiency. Nor any incompatibility between math and empiricism or math and physicalism. We do not need the natural laws of the world to be “written in the language of mathematics”. It’s enough, that we can speak math ourselves.

Discuss