A major part of scientism is the idea that maths and logic are not distinct from science, but rather that they arise from the same fundamental root — they are all attempts to find descriptions of the world around us. The axioms of maths and logic are thus equivalent to the laws of physics, being statements of deep regularities of how the world behaves that enable us to describe and model the world.
My article advocating that mathematics is a part of science was recently posted on Scientia Salon. This was followed by an article by Massimo Pigliucci which took the opposite line and criticised the return of “radical empiricism”.
In response I wrote about the roots of empiricism, defending the radical empiricism that Pigliucci rejects. That post was getting rather long, so I have hived off parts into this post where I return to the distinction between mathematics and science. This is essentially a third part to my above two posts, countering various criticisms made on Scientia Salon.
To summarise the above arguments in two sentences, my critics were saying: “Well no, mathematics is anything but studying physical objects. It is the study of abstract concepts”, whereas I was saying, “Yes, mathematics is the study of abstract concepts, abstract concepts that are about the behaviour of the physical world”.
I have argued that maths and logic and science are all part of the same ensemble, being ideas adopted to model the world. We do that modelling by looking for regularities in the way the world works, and we abstract those into concepts that we call “laws of physics” or “axioms of maths” or of logic. Thus axioms of maths and logic are just as much empirical statements about the behaviour of the world as laws of physics. In part one I discussed other possible origins of mathematical axioms, while in part two I discussed the fundamental basis of empirical enquiry.
That leaves several possible differences between maths and science, which I address here:
Necessary versus contingent
One possible distinction might be that the axioms of maths and logic are held to be universally true, and thus perhaps “necessary”, whereas much of physics is held to be locally contingent. My reply is that we don’t really know which axioms and laws are universal and which are not, since we only have a limited sampling of the universe. Can we really rule out the possibility that other regions of the universe, well beyond the observable horizon, operate on radically different logical and mathematical principles requiring radically different axioms? All we can really say is that maths and logic appear to be, in our limited experience, universal. But that statement also holds about many laws of physics, so this is not the basis for a distinction between maths and physics.
Another possible distinction might be between statements about concepts or patterns that apply to the world (such as maths) and statements of existence, that is, statements of contingent fact that particular entities exist. However, physics is not simply about lists of what exists, it also is about abstract concepts and patterns. The concepts of conservation of momentum or the second law of thermodynamics are good examples, but this is true about all laws of physics, since all laws are general abstractions of patterns.
The most common objection to my arguments on Scientia Salon was that, ok, we may grant that the axioms of maths did originally come from experience of the world, but since then a whole edifice of maths has been built on those foundations, and that edifice bears little relation to real-world behaviour, and nor do mathematicians concern themselves with real-world behaviour.
That reply already concedes most of my point, since the whole edifice of maths is tautologically tethered to the axioms. Thus, if the axioms are adopted owing to their correspondence to the world, then the edifice is still “about” worldly behaviour. Nevertheless, many mathematical constructions are not instantiated in the real world, so what is their status?
Recall that the laws of physics and axioms of maths and logic in the world-model are abstractions of deep regularities in the world. It follows that, from those abstractions, one can conceive of many more entities than are actually instantiated. That applies to both maths and physics. One can write down mathematical structures that don’t actually exist; but similarly, I am violating no physical law if I postulate possible planets in possible solar systems; and further it is easy to conceive of possible species that don’t exist and possible biochemical pathways in cells that are not part of the biochemistry that earthly life actually uses.
These all follow from the fact that the fundamental laws are abstractions about regularities, and the number of possible entities that are consistent with them is far greater than the number of entities that actually exist (at least in any finite volume of universe). Since this holds for sciences such as physics and biology as much as maths it is not a basis for distinguishing between maths and science.
But, by critics will reply, mathematicians spend their time working with mathematical constructions without caring whether those constructions exist, whereas physicists and biologists concentrate on those things that do exist. This is true, but what mathematicians are actually doing is not listing possible constructions for the sake of it, they are getting after deeper properties that apply to all constructions, and thus equally to the ones that do map to extant properties of the universe.
Physicists adopt the same approach. Cosmologists, for example, do not confine themselves to thinking about our universe alone, they consider the ensemble of all possible universes that are compatible with the basic laws. That way they better understand the basic laws, and their consequences, and thus better understand our universe and why it is like it is. Thus, a course on cosmology will review all the possible universes arising out of the Friedman equation, not just the one that we think we’re in.
Mathematicians might work with n-dimensional spaces (not just the n that gives our universe), but then so do physicists. Physicists also work with other-dimensional versions of physics as a way of better understanding physics, and thus of better understanding our universe. We can only properly understand the properties of our own universe, with three apparent dimensions of space, if we understand how physics would be in other dimensional spaces. Just for example, the question of why we have three apparent spatial dimensions can only be addressed in that wider context.
Which brings us to a wider point about what the brain is doing. The brain is there as a ready-reckoner device containing a good “world model”, available to be consulted to make real-time decisions. In order to better make decisions, it is continually using its world-model to run simulations of what might happen, thinking about “What if?” scenarios.
The way to run a What-if? simulation is to consider ranges of possibilities. In a trite example, a chess-playing computer considers all the possible moves an opponent might make. Similarly, the behaviour of other humans is capricious and varied, and thus in making day-to-day decisions our brains have to factor in the range of possible responses from other humans, and thus has to continually consider What-if? scenarios over a large range of possibilities.
This may be why fictional stories are of high interest to humans. Fictional stories are all What-if? scenarios that are consistent with the basic regularities of how the world works, but are about particular entities and events that are not actually instantiated. Similarly, science fiction takes the basic world model and then varies it a little. If it varied it too much it would be uninteresting to us, as too far from our concerns. But science fiction is essentially about human-like beings in a slightly different situation. Evaluating possible happening and outcomes over a range of plausible possibilities is what the brain evolved to do as a means of making decisions.
This story-telling approach is reminiscent of how both physicists and mathematicians work, since they are trying to invent ideas that explain reality. The fact that mathematicians are interested in their constructed ideas for their own sake doesn’t place them in a different category. So are theoretical physicists (just look at string theory for example), and so are story-tellers. A novelist would be puzzled if you complained that the novel’s events had not actually happened, and yet every story ever told has been heavily rooted in human real-world experience and is still “about” human real-world experience. Maths is similar, all of it being rooted in the real world, regardless of whether particular constructs are instantiated.
So is there a difference between maths and physics?
In one sense maths is very different from physics. A mathematician will commonly take a set of axioms as a given, and then deduce theorems deriving from those axioms. That mathematician (if a pure mathematician and not an applied mathematician) may not ask about and might not be concerned with the real-world correspondence of the theorem.
A theoretical physicist might similarly take a set of laws as a given, and then deduce the behaviour of a physical system from those laws, but any physicist would be concerned with the real-world implementation of the results.
Part of the difference could be that the mathematician is usually sufficiently confident in the real-world correctness of the axioms that the real-world truth of the resulting theorem is a non-issue. If the axioms are real-world true and provided there is no mathematical mistake, then the resulting construction will be real-world true.
In contrast, the physicist will be concerned to test the products, since physics is more complicated than maths (in the sense of messy, contingent, real-world complications, and in the same sense that biology is then way more complicated than physics) and thus the physicist will have much less confidence in the laws and methods used.
Where mathematicians do invent new areas of mathematics and thus adopt new axioms, they are indeed concerned about the real-world correspondence of the axioms, and indeed new areas of maths are usually inspired by real-world issues (an example being paraconsistent logic, which is an attempt to deal formally with the real-world scenario of having inconsistent information).
On occasions where the real-world truth of the axioms is in doubt, this is a matter of discussion among mathematicians. For example, the Axiom of Choice is clearly real-world true for finite sets, but gives results that don’t seem to be real-world true when applied to infinite sets (for example the Banach–Tarski paradox). Thus the validity of the Axiom of Choice applied to infinite sets is debated by mathematicians, to the extent that they commnly consider three sets of axioms, namely the ZF axioms, ZF plus the Axiom of Choice, and ZF plus the negation of the Axiom of Choice.
Thus one can indeed point to differences between mathematics and physics. One can certainly distinguish between activity based on proximate and current observation of the world, and activity which just uses the stored empiricism now distilled and encoded into axioms and laws. But that doesn’t alter the fact that, ultimately, all of the activity derives from contact with the empirical world.
The distinction between, on the one hand, using a combination of proximate empiricism and stored empiricism, and, on the other hand, using only the stored empiricism, is not one about the fundamental origins of the knowledge, it is mostly a matter of pragmatics.
Further, that distinction does not distinguish neatly between maths and science, since the use of stored empiricism is ubiquitous in all areas of human activity — that being the entire point of the brain, that we can know things without having to directly observe them.
Instead, the differences between maths and physics seem more to be matters of style, rather than basic epistemology. There are bigger differences in style across the sciences (say between primatologists and theoretical cosmologists) than there are between mathematicans and theoretical physicists (with string theorists straddling that border). Hence — while not denying interesting and significant differences of style across the different areas of knowledge — in terms of fundamental epistemology it makes sense to regard science, mathematics and indeed logic as a unified whole, different parts of an ensemble aimed at modelling the world.