Defending scientism: mathematics is a part of science


While the term “scientism” is often a rebuke to those considered to be overstepping the proper boundaries of science, plenty of scientists will plead guilty to the charge so long as they get a say in how the term is defined. The “scientism” that I defend is the claim that, as far as we can tell, all human knowledge is empirical, deriving from empirical contact with reality. Further, that empirical reality seems, as far as we can tell, to be a unified whole, and thus our knowledge of reality is also unified across different subject areas so that transitions between subjects are seamless.

What we call “science” is the set of methods that we have found, empirically, to be the best for gaining knowledge about the universe, and the same toolkit and the same basic ideas about evidence work in all subject areas. Thus there are no “other ways of knowing”, no demarcation lines across which science cannot tread, no “non-overlapping magisteria”.

A related but different stance is expounded by philosopher Massimo Pigliucci in his critique of scientism [1]. Pigliucci instead prefers the umbrella term “scientia”, which includes “science, philosophy, mathematics and logic”. This sees mathematics and logic as epistemologically distinct from science. Indeed Pigliucci has remarked:

it should be uncontroversial (although it actually isn’t) that the kind of attention to empirical evidence, theory construction, and the relation between the two that characterizes science is “distinctive enough” … to allow us to meaningfully speak of an activity that we call science as sufficiently distinct from … mathematics.

Mathematics is a huge area of knowledge where science has absolutely nothing to say, zip …” [2]

In this piece I argue that mathematics is a part of science. I should clarify that I am taking a broad interpretation of science. Nobody who defends scientism envisages science narrowly, as limited only to what is done in university science departments. Rather, science is conceived broadly as our best attempt to make sense of the empirical evidence we have about the world around us. The “scientific method” is not an axiomatic assumption of science, rather it is itself the product of science, of trying to figure out the world, and is now adopted because it has been found to work.

I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is:

1 + 1 = 2.

Do you accept that statement as true? If so (and here I presume that you answered yes), then why?

I argue that we accept that statement as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world. [3]

It is a fact about the world in the same way that apples falling downwards are a fact about the world. There is no good reason to place these two different facts (gravity and maths) into two incommensurate domains of knowledge. Our understanding of both derives from empirical reality, and thus both are equally “scientific”.

Having asserted that, let me argue against possible alternative answers to my question of why we accept that 1 + 1 = 2.

Maths is derived from axioms

One answer — perhaps the commonest — asserts that maths is not derived from empirical observation but instead derives from axioms. You might assert that you accept 1 + 1 = 2 because it is proven so from the basic axioms of maths. You might point to Peano’s axioms and assert that from those one can logically arrive at 1 + 1 = 2. [4]

My first reply is that I don’t believe you. I don’t believe that there was a time in your life when you were dubious about the assertion 1 + 1 = 2, but then consulted Peano’s axioms, and after some logical thought concluded that, yes, 1 plus 1 really must equal 2. I assert that you accepted 1 + 1 = 2 long before you knew anything about Peano’s axioms, and that you accepted it because it works in the real world: if you had two sweets you could give one to your pal and eat the other yourself. [5]

But, even if your belief that 1 + 1 = 2 does derive from axioms, whence your faith in those particular axioms? How and why did Signor Peano arrive at that set of axioms? I assert that they were arrived at with the fact of 1 + 1 equalling 2 being a necessary consequence. Had Peano’s axioms resulted in 1 + 1 equalling anything other than 2 then the axioms would have been rejected as faulty. Signor Peano would have been told to go away and come up with axioms that worked (ones compatible with the non-negotiable truth that 1 + 1 really does equal 2).

Thus, the axioms mathematicians adopt are not arbitrary, chosen by whim or fiat, they are chosen to model the empirical world. Mathematics is thus distilled empiricism. The same can be said about logic and reason. In order to get from Peano’s axioms to derived results you need to use logical reasoning. What validates that logic and that reasoning? Again, I assert that empirical reality validates them. The reason that we adopt logical axioms such as the law of non-contradiction is that they hold in the empirical world. [6] How else would we know which logical axioms to adopt? Thus the whole edifice of mathematics and logic is a distillation of empiricism, created and developed as a model of the basics of how our world works.

Mathematics is arbitrary

Nevertheless, some might assert that no, mathematics is a self-contained logical system entirely distinct from empirical reality, and that any correspondence between mathematics and science is simply a coincidence. Some might even assert this with a straight face. It leads to puzzlement over what Eugene Wigner called “the unreasonable effectiveness of mathematics” when applied to science, but there is no puzzle if mathematics describes deep properties of our empirical universe and is derived from that universe. The idea that mathematics is arbitrary and independent of our universe would be more convincing if mathematicians spent as much time pursuing maths based on 1 + 1 equalling six and a half as they do with 1 + 1 = 2.

A more sophisticated version of this answer accepts that mathematics originally derived from our universe (with, for example, Pythagoras’s theorem resulting from drawing on bits of paper, or from attempts to get a building’s walls square), but points out that nowadays mathematicians experiment with all sorts of axioms that are not first suggested by observation.

As an example, consider the generalisation of the “flat” geometry developed by Euclid to the “curved” geometries developed by Carl Gauss, Bernhard Riemann and others. The relaxing of the parallel-line postulate of Euclid to produce non-Euclidean geometries was not motivated by observations but by thinking about the structure of the axiomatic system. Surely this is a non-empirical approach that distinguishes mathematics from science?

Well no. Theoretical physicists do this sort of thing just as much as mathematicians. They take their set of empirically derived axioms (though in physics these tend to be called “laws” rather than “axioms”) and think about them; they experiment with different axioms/laws and work out the consequences. Often they are not immediately motivated by a match to observations but are following their intuition.

They are still, though, working with an axiomatic system that is essentially distilled from the empirical universe, and they are using an intuition that is also very much a product of the empirical universe. Curved geometry — developed by the mathematician Riemann — was later found to be useful in describing the universe when the physicist Einstein — also following a path of logic and intuition — developed the theory of General Relativity. If anyone wants to draw a demarcation line between domains of knowledge, the line would not be between the mathematician Riemann and the physicist Einstein.

Why is it that mathematicians’ intuitions so often produce mathematics that is later found to be useful to physicists? I argue that their experimentations with axioms are productive because their logic and intuitions are also empirical products. Thus a mathematician has a good idea of which changes to axioms are sensible and which are not. Allowing parallel lines to diverge (and thus producing non-Euclidean geometry) is sensible; adopting “one plus one equals six and a half” is not. [7] In both mathematics and physics, if the experimentation produces results that are nonsensical when compared to our universe then they will not be pursued. The empirical universe is in both cases the ultimate arbiter.

At the cutting edge it can, of course, be unclear whether maths and/or physical theories “work”. A current example is string theory, where a generation of theorists is exploring the mathematics of strings. Maybe it’ll lead to new physical theories unifying quantum mechanics and gravity, and maybe not. At the moment, though, one could not really say whether string theory was “mathematics” or “theoretical physics”. This emphasises the seamless transition between those fields, with string theory straddling the (arbitrary and unmarked) boundary.

An aside before proceeding. Gödel’s incompleteness theorem tells us that even if we have a set of axioms such as Peano’s axioms, which underpin the natural counting numbers and which yield the statement that 1 + 1 = 2, there will be other statements about the natural numbers which are true, but which cannot be shown to be true from the axioms. A further result tells us that the axioms cannot be used to show that the system built from those axioms is consistent. This fundamental limitation of an axiom-based approach shattered hopes of mathematics ever being a complete, consistent, self-validating and self-contained system.

From a scientific point of view, with mathematics being seen as a part of science, such limitations are unsurprising. Science is derived from empirical evidence and our available evidence will always be a small and incomplete sample of the universe, and thus scientific results are always provisional, in principle open to revision given better data.

But mathematicians spend their time exploring axioms that may be unrelated to the empirical world

This is true. But then so do many theoretical physicists (ask any string theorist)! Such a mathematician may not care about correspondence with reality, but is simply exploring the results of different axioms for fun.

If we ask what the human brain evolved for, it evolved to assimilate sensory data, to model those data, to perform deductive reasoning on those data, and to run “what if?” simulations as an aide to decision making. E.g. “If I do X how will others react?”. Such “what if?” simulations are based on empirical reality, but then simulate possible variations on that reality. For example, all humans like stories, and a novel can be considered a “what if?” alternative reality, in which the characters and events are not “real”, but are the sort of things that could be real.

Mathematics is like this. It is ultimately based on truths that we arrived at empirically. But then mathematicians explore the consequences of those truths, and perform “what if?” simulations in which they consider alternative realities. For example they may think about 5-dimensional space rather than the usual 3-dimensional space. This mathematics is “about” the empirical world in the same way that Jane Austen’s Pride and Prejudice is “about” the real world.

That is not to say that a 5-dimensional orthoplex is a “real” object any more than Mr. Darcy is a real person, but the whole enterprise is still very grounded in the empirical world. If an axiom or an avenue of mathematics becomes nonsensical in comparison to the empirical world then it is simply not pursued and is regarded as uninteresting. The fact that many “abstract” areas of mathematics have later been found to have useful physical applications just shows how good mathemiticans and their empirically grounded intuitions are at judging “what if?” scenarios.

Our maths is the product of pure logic, deriving only from human intuition

Many will disagree with me and assert that human intuition is a primary source of knowledge, distinct and separate from empirical evidence. Indeed this idea is popular with some philosophers, who argue that introspection and thought are the wellsprings of their philosophical knowledge, and thus that philosophy is a domain distinct from the empirical domain of science. [8]

However, what basis do we have for supposing that our human intuition produces accurate knowledge about the universe? The first reason is that our intuition has been developed and honed over our lives based on our sense data about the world around us. Thus our intuition is very much an empirical product.

Further, we can ask about instinct (that portion of our intuition that is not the product of life experiences, but is encoded in the genes). Our genetic programming will also be a product of empirical reality. Our brains are the product of evolutionary natural selection, and thus have developed to make real-time decisions that aid survival and reproduction. Obviously decision-making that bore no relation to the real world would be useless, and thus we can have some confidence that our intuitions are to a large extent programmed to produce decisions well-aligned to empirical reality.

Of course natural selection is not a perfect programmer, and anyhow is not aiming at a perfect and unbiased decision-maker, it is aiming at the one best at survival and reproduction. Thus we would expect our intuition to be reliable only with respect to the everyday world relevant to survival and reproduction, and to be unreliable about aspects of the universe (such as quantum mechanics and general relativity), that are irrelevant for everyday life.

We’d thus expect our intuition to be a folk metaphysic, good enough for many purposes, but full of biases and foibles, particularly so where an inaccurate assessment might actually aid survival and reproduction. An over-active pattern-recognition detector and the Lake Wobegone effect are likely examples of this. Visual illusions such as the checker-shadow illusion show how easily the human intuition is fooled, in this case precisely because the human intuition is making some assumptions about how the world works, and thus about lighting and shading. [9]

A critic might, though, accept that some of our intuitions are related to empirical reality, but then argue that intuition also gives access to knowledge that is not empirical and cannot be arrived at by empirical means. My response is to ask what basis the critic has for that assertion and what reason he has for supposing that “non-empirical knowledge” has any reliability or validity.

From the evolutionary perspective we have no good reason to suppose that intuition is anything other than an imperfectly and empirically programmed device that models the empirical world — after all, failing to find enough to eat, ending up eaten by a predator, or finding a mate and successfully rearing children, are all aspects of a brute empirical world. Thus we should accept intuition as a useful “quick guide to reality”, but ultimately we should not accept human intuition except where corroborated by empirical evidence. Indeed, the whole point of the scientific method is to use empirical evidence to do much better than just consulting our “quick guide” intuition. [10]

Our maths is the only possibility

The last alternative answer that a critic might advance [11] is that we accept the claim that 1 + 1 = 2 because it must be true, it is the only logical possibility. Thus, such a critic will say, 1 + 1 equalling six and a half is simply nonsensical. Such a person would not merely be asserting that it is impossible in our world, but that it is impossible in all possible alternative worlds.

Do we know this? And, if so, how? Has anyone given a logical proof of the impossibility of such an alternative scheme? Any such proof could not use any axiom or logic derived from or validated by our empirical world (that would only show that such alternatives did not occur within our world). But without that, how would one go about showing that the logic of our world is the only one possible?

One could not use our-world logic for such a task and nor could we use human intuition, since our intuition is very much derived from and steeped in the logic of our own empirical world — indeed our brains have evolved precisely to model the logic of our world — and thus we would not expect them to be in any way useful for contemplating radically different alternatives.

But, even if we were to grant the claim that our world’s logic is the only possible system of logic, that would still leave the question of how we came to learn about that logic. And the only plausible answer is that we learned from observation of the empirical universe and thence deduction about the logic by which it operates.

Further, even if mathematics is “necessarily true” that does not necessarily distinguish it from empirical science. For all we know, laws of physics might be “necessarily true” in the same way. At a deep level a theoretical-physics description of fundamental particles and the properties of space and time is a mathematical description. [12] While some philosophers distinguish “necessary” mathematical truths from “empirical” facts about science, this distinction seems less and less appropriate as physics develops. Pigliucci states that:

There are mathematical explanations for why certain things are impossible (like crossing all the bridges of the town of Königsberg exactly once) which trump, or make superfluous, or are more basic than, any scientific (i.e., empirical) explanation.”

Pigliucci references an article by Marc Lange [13] which distinguishes “distinctively mathematical” explanations from “causal” scientific explanations. Yet such “distinctively mathematical” explanations are commonly used in science, a typical explanation being: “because the laws of physics are invariant under the Lorentz transformation” (where the Lorzenz transformation is mathematical).

Or take the question: “why does there not exist a particle consisting of three up-quarks with a spin of 1/2 and a mass similar to that of the proton?”. The answer is that there is no way of making such a combination have a wavefunction that is anti-symmetric under exchange of any two particles. This is a “distinctively mathematical” explanation of exactly the bridges-of-Königsberg type, but is also the sort of explanation that fundamental theoretical physics increasingly arrives at. The explanation is not in terms of causes but in terms of deep axioms/principles/laws, such as the need for a wavefunction to have a definite symmetry.

But, if we ask further why such a particle can’t have a mixed symmetry we can only resort to the empirical fact that they don’t occur in nature. Similarly, a “distinctively mathematical” explanation for why one cannot cross the bridges of Königsberg exactly once relies on axioms for which the only ultimate validation is empirical.


I have argued that all human knowledge is empirical and that there are no “other ways of knowing”. Further, our knowledge is a unified and seamless sphere, reflecting (as best we can discern) the unified and seamless nature of reality. I am not, however, asserting that there are no differences at all between different subject areas. Different subjects have their own styles, in a pragmatic response to what is appropriate and practicable in different areas. For example, a lab-based experimental science like chemistry has a very different style to an observational science like astronomy. [14] Further, biochemists studying detailed molecular pathways in a cell will have a very different style to primatologists studying social interactions in a wild chimpanzee troop.

Such differences in style, however, do not mandate that one of those subjects be included within “science” and another excluded. The transitions in style will be gradual and seamless as one moves from one subject area to another, and fundamentally the same basic rules of evidence apply throughout. From that perspective mathematics is a branch of science, in the same way that so is theoretical physics. Indeed, some theoretical physics is closely akin to pure maths, and certainly far closer to it in style and content than to, say, biochemistry. The different subject labels can be useful, but there are no dividing lines marking the borders. No biochemist worries about whether she is doing biology or chemistry, and string theorists don’t worry much whether they are doing maths or physics.

Thus, in arguing that a subject lies within the broad-encompass of “science”, one is not asserting that it is identical in style to some branch of the generally-accepted natural sciences, but that it belongs to a broad grouping that spans from studying molecules in a chemistry laboratory, to studying the social hierarchies of a baboon troop, to theoretical modelling of the origin of the universe, and that it belongs in that group because epistemologically the knowledge has the same empirical source.

I thus see no good reason for the claim that mathematics is a fundamentally different domain to science, with a clear epistemological demarcation between them. This same set of arguments applies to the fields of reason and logic, and indeed anything based on human intuition. All of these seem to me to belong with science, and all derive from the same source, our empirical experience of the universe and our attempts to make sense of it.


[1] Massimo Pigliucci, Midwest Studies in Philosophy 37 (1):142-153 (2013) “New Atheism and the Scientistic Turn in the Atheism Movement”.

[2] See Pigliucci’s article Staking positions amongst the varieties of scientism.

[3] A pedant might point out that in modular arithmetic, modulo 2, 1 plus 1 would equal 0. I am taking 1 + 1 = 2 to refer to simple counting numbers; one apple plus one apple equals two apples. If we ask further about the basic concepts of “1”, “2”, “+” and “=” I would again base them on patterns discerned in the empricial world, which is of course how humans first came up with those concepts.

[4] Giuseppe Peano, 1889. Arithmetices principia, nova methodo exposita. Wikipedia account

[5] A pedant might point out that that equates to 2 – 1 = 1, not to 1 + 1 = 2.

[6] Indeed the great Islamic polymath Avicenna wrote, c AD 1000, that: “Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned”, a direct derivation of logic from empirical experience!

[7] The Axiom of Choice is an example of an axiom adopted largely because it feels intuitively right to mathematicians, plus they like the results that it leads to.

[8] There is a vast philosophical literature on this issue, with Kant’s Critique of Pure Reason being influential.

[9] For the Lake Wobegone effect see the this wikipedia article. For the checker shadow illusion see here.

[10] An obvious example being the need for double-blinding in medical trials, which originated from the realisation of how unreliable human intuition, based on anecdotes and a partial memory, actually is.

[11] Unless you have other alternative answers?

[12] See Tegmark, Max, 2014 “Our mathematical Universe” for pushing this argument all the way.

[13] Marc Lange, 2012, What Makes a Scientific Explanation Distinctively Mathematical?

[14] One should ignore commentators who over-interpret overly-simplistic accounts of the “scientific method” and claim that only lab-based experimental science counts as science.