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December 11 2025

This is a follow-up to my posts Academic philosophy: my quixotic quest and A critique of philosophical objectivity. The latter post argued against the notion of “objective truth,” that is, truth independent of humans in some crucial sense. I think it’s anthropomorphic to suggest that the nonhuman world makes our human-created, human-centric representations of the world true or false, independently of us.

In addition to logic, which I discussed and criticized a bit in my first post, mathematics is considered by many philosophers to be a paragon of objective truth. Surely the fact that 1 + 1 = 2 does not depend on humans, right? On the other hand, everyone would admit that 1, +, =, and 2 are all human symbols. Indeed there are many such human symbols. Corresponding to the Arabic numerals 1, 2, 3, 4, 5 are the Roman numerals I, II, III, IV, V; in English one, two, three, four, five; in Spanish uno, dos, tres, quatro, cinco. We standardly use a base-10 number system with numerals 0 through 9, but that’s arbitrary, and there are alternative bases, used especially in computer programing, for example base-16, or hexadecimal, in which the letters A through F come after 0 through 9, so that the base-10 numbers 15 and 16 are represented by F and 10 in base-16. There’s also base-2, or binary, where the only numerals are 0 and 1. By the way, in base-16 it’s still true that 1 + 1 = 2, but 5 + 5 = A. And in base-2, 1 + 1 = 10.

The foundation of math, I would say, is counting and measurement in the physical world, space and time. Without such practical application, “pure” mathematics would be nothing more than a game, an intellectual amusement. Conversely, without mathematics, our descriptions of the world would be deeply impoverished and unproductive. Math is so useful that it allows us in many cases to predict the future and extrapolate the past with great accuracy. In this sense, math conjoins the arbitrary and the non-arbitrary: we humans created our mathematical symbolism, but for the most part we did not create the regularities in nature that bring math to life and make it shine.

Math is, from my perspective, an activity of humans and possibly other conscious, living beings. We say that the “natural” numbers are 0, 1, 2, 3… (though some historical definitions did not include zero), and of course it’s natural for us to count. What bothers me about philosophical interpretations of math, the alleged objective truth of math, is that objective truths are supposed to be essentially independent of humans. Thus, on this conception, the existence of the natural numbers would not depend on the existence of counting or humans or living beings. The natural numbers are said to be an ordered series or set, but the “order” in question is in neither space nor time; rather, the claim is that objective mathematical truths are nonphysical, timeless, not connected with any particular points in space and time.

To me, this is nonsense. It’s anthropomorphism, projecting our human activities such as counting onto the nonhuman universe. Consider one of the simplest mathematical concepts: addition. Philosophers claim that 1 + 1 = 2 is an objective truth, and moreover an abstract, nonphysical truth. But what in the world is spaceless, timeless addition? I have one dollar bill, then someone gives me another dollar bill, so I have two dollars. $1 + $1 = $2. That makes sense. I go to the grocery store, put one apple in my cart, then another apple. 1 apple + 1 apple = 2 apples. At the checkout, I pay my $2 for the 2 apples (inflation, you know), leaving me broke again, because $2 - $2 = $0. These exchanges, these actions, these mathematical operations, addition and subtraction, occur in the physical world, essentially involving space and time. I think it’s fair to say that 1 + 1 = 2 is an abstraction: for all things, 1 thing + 1 thing = 2 things. Yet this abstraction doesn’t make 1 itself a thing, nor 2 itself, existing apart from all other things. Nor does it imply the existence of (0,1,2,3…), the natural numbers, independent of humans and the physical universe.

I said earlier that math conjoins the arbitrary and the non-arbitrary, and I think the notion of objective mathematical truth ignores the arbitrary part of math. For comparison, consider the arbitrariness of spelling. The word “apple” is spelled a-p-p-l-e, but that truth is not independent of humans; we simply decided to spell the word that way. We also decided to use base-10 and count 1, 2, 3, etc. Given our basic mathematical symbolism, mathematicians can prove many interesting, surprising facts about numbers, for instance about prime numbers. Nonetheless, I find it bizarre to call 1 + 1 = 2 an objective truth, independent of us. On the contrary, I would say that it depends entirely on us. That’s just how counting works. Consider any randomly chosen number XN, where X is a finite series of decimal digits of nonzero length, and N is also a decimal digit. What is XN + 1? We know that if N < 9, then XN + 1 = X(N+1). In other words, if N is 3, then X3 + 1 = X4. Whereas if N = 9, then X9 + 1 = (X+1)0. And this is neither a “discovery” that we somehow have to prove, nor is it an “objective” truth. Rather, it’s simply how counting works. If someone claimed that X3 + 1 = X5, we’d call them daft, perhaps insane. Such a person would have failed to learn how to count, just as a person who wrote “appel” would have failed to learn how to spell the word. This is illustrated hilariously by the film “Monty Python and the Holy Grail” in the Holy Hand Grenade scene. After consulting the unambiguous, tedious instructions from Book of Armaments, the inattentive King Arthur, directed toward the number three, pulls the pin of the grenade and counts, “one, two, five!”

Let us (as if you had a choice) follow another analogy: tennis. There are both arbitrary and non-arbitrary aspects to tennis. The rules of tennis are arbitrary, invented by humans. The scoring in tennis involves a different, bizarre counting scheme, unlike any other. A tennis match is comprised of points, games, and sets. That’s why they say, “game, set, match” when someone wins the final point. In order to win a game, a player must win at least 4 points in that game and also outscore their opponent by at least 2 points. But they don’t count points as 0, 1, 2, 3, 4. Instead, they say “love” for zero. Don’t you love that? Every game starts at love, love. So much love to go around! If a player at love wins a point, their score is now—I kid you not—15. The tennis numbers, which we can hardly call “natural,” are love, 15, 30, 40. Although if both players have 3 points, they don’t say the score is 40-40 but rather “deuce.” Again, I’m 100% serious, not joking! If the server wins the point after deuce, the score is now “ad-in,” while if the returner wins the point, the score is “ad-out,” where “ad” is short for “advantage.” If a player loses the advantage by losing the next point, the score returns to deuce.

Although humans decided on tennis scoring, arbitrarily, humans cannot just decide to hit a tennis ball at 500 kilometers per hour or make the tennis ball fly in a square wave. We make the rules of tennis but not the laws of nature, and tennis players must obey both. Tennis is not an abstract game; it’s concrete, sometimes played literally on top of concrete. Like tennis, math is also an activity constrained essentially by physics. And nobody would consider the rules of tennis to be timeless, objective truths, so I’m not sure why philosophers consider the rules of math to be timeless, objective truths. In both cases, the rules of the “game” would be pointless, make no sense without a physical instantiation, without active human participation. And in both cases, the result of the activity is somewhat unpredictable, which makes the activities interesting. This is not to say that 1 + 1 will ever add up to 3 or that a tennis player will ever proceed directly from 15 to 40. The point, if I may pun, is that the outcome of a tennis match is not predetermined, and the outcome of applying math to science, for example, is not predetermined either. The laws of nature, the mathematical equations representing them, cannot be derived by logical inference from axioms. The math must fit the empirical data observed in the world, and when scientists happen to discover a match (more puns!), the outcome is often surprising, even startling.

It’s true that we couldn’t do much science without mathematics. On the other hand, we couldn’t do much science without natural language either. Indeed, we couldn’t do much math without natural language. Try teaching a child math without ever teaching them any other language. I suspect that you’d have great difficulty reaching something as basic and simple as 1 + 1 = 2. (Actually, don’t try that, because it would be extremely unethical. But you can try it as a thought experiment, in your evil mind.) In my earlier blog post, I explained why the existence of objective truths independent of us is not entailed by the fact that language is crucial to us in describing the world. Likewise, the fact that math is similarly crucial does not entail the existence of objective mathematical truths independent of us.

Sometimes it’s claimed that mathematics is the language of nature, but I’m skeptical. Humans measure, count, add, subtract, multiply, divide. Not only do we divide in the mathematical sense, we also divide the world into units, categories, and objects. These divisions are, at least to an extent, arbitrary. I’ll repeat a couple of quotes from my previous blog post:

Our vision divides the world into a relatively small number of objects, small enough that we can reasonably formulate plans of action regarding those objects: eat this, run away from that.

Much of our representation, including much of science, is simplification, cutting the world into bite-sized chunks for easy digestion.

In what sense does the nonliving world count things, add things, divide things, etc., and what exactly are these things? Where, when, and why does “natural” mathematics occur, mathematics that is independent of us? If the answer is nowhere and at no time, outside space and time, I would wonder whether the “why” is absent too, if there’s any reason for us in space and time to believe in abstract mathematics and moreover the relevance and applicability of abstract mathematics to space and time. How is it even connected to us? I suspect that there’s a philosophical, anthropomorphic, sometimes religious temptation to claim that the mathematical laws of nature are not just a scientific description of how the universe works but rather the essential cause of how the universe works. In other words, the reality that we observe would be a physical instantiation of the abstract laws, perhaps written by God.

I think of mathematics and science as reliable tools. The reliability of the tools certainly tells us something about the world, but that doesn’t mean the tools are somehow entirely independent of the toolmakers. Scientists are engaged in a kind of mapmaking. But making a map of the world doesn’t presume or require that the world itself was made from a map! Thus, the only touchstone for a map is the judgment of the map makers and users, whether the map serves their purposes. The goal is not fidelity to some original map, as if the area to be mapped were merely an intermediary between one map and another map. However, philosophers seem to aspire to reconstruct some kind blueprint for the world, a set of instructions as it were, where the blueprint already takes a kind of representational form, resembling language or symbolism, which would make it possible to judge our human descriptions and representations of the world against the blueprint, thereby providing the alleged objectivity of truth. Again, I think this is unjustified anthropomorphism.

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