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.

### Story telling

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.

Paul BratermanWhat is the fact in the world that corresponds to the well-known mathematical result that square root 2 is irrational? Or, if there is no such fact, how does your thesis survive?

CoelPost authorHi Paul,

The short answer is that the “fact in the world” that corresponds to the square root of 2 being irrational is the fact that the square root of 2 is irrational, which is just as much a physical statement about how the world is as a mathematical statement.

The longer answer is that all physical descriptions (aka “facts”) are abstracted models of reality. Nothing in science is a pure model-free “factual” observation. Our sense organs give us photon-arrival-event information, and everything else is a model built on that.

For example, the physical concept “momentum” is an abstraction that is not observed directly, but is a concept that helps model observations. We can say that it is an observed fact that “momentum is conserved” in the world, but that’s a pretty high-level abstraction. Yet, no-one disputes that this is “physics”.

Concepts such as counting numbers and multiplication and division are also abstracted models of real-world behaviour in the same way that “momentum” and “electromagnetic charge” are. Thus, the square root of 2 being irrational is a “fact in the world” in the same sense that “momentum is conserved” is a “fact in the world”.

Hugh JidietteHi Coel,

First time poster, and fan of your blog. Quick comment: someone like Kant would agree that everything starts with experience, yet say that we know things a priori. The difference between a priori and a posteriori/empirical would be that a priori knowledge is knowledge independent of any particular experience, whereas a posteriori knowledge requires particular experiences. E.g., we would need particular experiences to learn Newton’s inverse square law of gravity, but we could learn 1+1=2 from general experience.

CoelPost authorHi Hugh,

Is

a prioriknowledge “from general experience” (as opposed to particular experience), or is it more things that are claimed to be known entirely independently of experience? My counter would be to claim that nothing is known entirely independently of experience, and that all knowledge ultimately derives from experience.Hugh JidietteI, and Kant, I take it, would agree that you can’t know anything entirely independent of experience, because a person born without any senses would probably be unable to form any judgments whatsoever; he’d be more like a vegetable.

On Kant’s view, to derive Newton’s laws of gravity, we would need the planets (for example) to move in a very particular way. But a priori knowledge is the type gained from any old experience; there isn’t a particular experience–like in the Newton example–that is needed to learn 1+1=2. We necessarily make judgments about the world with mathematics like 1+1=2. It’s hard to see how any experience could falsify 1+1=2, though I take it you disagree with that.

CoelPost authorHi Hugh,

It’s hard to imagine experience falsifying 1+1=2 exactly because that is the way that our world works, and all our experience and indeed our intuition and our imagination are steeped in that world and evolved out of that world.

verbosestoicWell, all knowledge can’t ultimately derive from experience because we need to know certain things — or at least possess certain concepts — before we can make any sense of experience. Like differentiation, for example. There’s no way to get the idea that things are different from experience unless you already know and have a means of determining that things are different. Once you have that, you can get similarity, but you can’t go the other way around.

A bit of a sidebar, though, because what I really want to comment on is your attempt to demonstrate that mathematics and logic are, in fact, ultimately just as much ways to model the world and depend as much on it as science does. The issue is that you are focusing on the pragmatic use of it rather than what the fields themselves actually do. So let’s think about what happens inside the fields themselves and ask: How do the fields determine if a work of mathematics or a work of logic is valid and valuable to the field?

Do they do that by checking it against the world and seeing how well it models the real world? No, and I think that most mathematicians and logicians would be very upset with you if you insisted either that they did or that they ought to. Mathematics and logic are of interest to those fields not because they better model the real world, but because they lead to interesting mathematical and logical systems. Some of the most interesting mathematics and logics simply couldn’t be used to model the world at all. The interest of mathematics and logic to OTHER fields might be that, but that’s not what’s interesting to mathematicians and logicians (generally).

So can we say that what makes a mathematical or logical system interesting is how well it can be used by other fields to model the world? This fails for the same reason. Mathematicians and logicians might think it nice if this can be done, but the systems would be still as interesting and valid inside those fields if no one outside of that field could possibly use them as a tool to better model the world. So again the validity and value of mathematics and logic is not determined by how will it can be used as a tool to model the world.

Because what is a valid and valuable mathematical or logical system is not determined by how well it can be used to model the world, these fields are not there to model the world or help us to model the world. So mathematical and logical systems are not JUSTIFIED by how well they model or help us to model the world. Therefore, they are not justified empirically, where empirically means “By how well they model the world”. Therefore, they are not any kind of empirical science. And, therefore, you have an exception to your scientism claims.

CoelPost authorHi Verbose Stoic,

True, but I would argue that the most basic ability to learn is the result of evolutionary programming. And that programming is, just as much, the result of experience of the world around us (though it is natural selection over eons, rather than our own personal experience).

Agreed, they don’t. What they do is check that it correctly follows from axioms. However, if those axioms are themselves the product of experience of the world (distilled empiricism) then the resulting model is still, ultimately, derived from experience of the world.

verbosestoicWhich wouldn’t be experience in the way needed to justify a claim of it being empirical, which is what you want. At this point, you’d be trying to prove your claim using terminology that no one arguing against you uses. This is one of the prime philosophical sins, and yet the most common: redefining your terms so that they avoid challenges, but in the process expanding them so much that they are meaningless, and also don’t address the concerns your opponents have.

“Derived from experience of the world” can be a pretty broad notion. After all, it can be argued that without having at least some experiences we can’t think at all, and so everything is derived from experience of the world. But that’s not what you need. This is why I focused on JUSTIFIED by experience of the world, and asked if we determine what is a valid mathematical theory by appealing to the world. And even with the axioms, we don’t. Consider these two axioms:

A1: A right angle shall be defined as an angle of 90 degrees.

A2: A right angle shall be defined as an angle of 45 degrees.

Both of these are perfectly valid mathematical axioms, and can lead to perfectly valid mathematical theories and systems. Only A1 could possibly be justified or even derived from real world experience, and even there it doesn’t have to be in order to be valid. As such, mathematical axioms and systems are not justified by real world experience, or anything about the real world. It’s only if you try to extend things out that you even get to any kind of credible argument that they might be … but at that point, your extended definition is not addressing the point your opponents are making.

CoelPost authorI agree that in treating evolutionary programming as “experience of the world” I’m making it a rather broader concept than normal. My argument, though, is that such evolutionary programming underpins all empirical learning, and thus it is a reasonable extension. Afterall, empirical data alone don’t do anything. You can direct empirical data at a house brick and it will not learn anything from it. We gain information empirically only because we have a programmed ability to learn, which is itself the result of past experience of the world.

I still argue that they are. The whole point is that mathematicians do not spend their time working with mathematical systems built from mathematical axioms that are nonsensical in the real world, and which are directly contrary to the real world. In principle it might be possible to do that, but in practice people only do it, if at all, to a limited extent, as a “story-telling” variation on the real world.

verbosestoicSomehow, I screwed up the blockquoting on that one, so it’s a little confusing. But I can’t edit it to fix it up …

CoelPost authorI’ve fixed it.

verbosestoicIt’s not a reasonable extension because it ends up trying to beat your opponents by definition and not by argument. If we take the idea of innate ideas and a priori justification, a lot of that is based on the idea that we can’t learn all important concepts by our OWN direct empirical experience because we need to have certain ideas and certain things justified first. What you do here is take that argument, accept it … and then define that as being empirical ANYWAY because it’s needed for us to learn directly from our own empirical experiences. This would be, at best, using empirical in a completely different sense than your opponents and declaring victory over them, by adapting their own argument to suit your needs. Essentially, you’d accept their argument which should prove them correct, but instead you alter your argument and definitions to make your own argument work, to be able to say that you’re still right despite accepting, essentially, that THEY are right. That’s … not really a response [grin].

Non-Euclidean geometries would disagree with you, there. For the most part, mathematical systems do not, in any way, have to relate to the real world in order to be considered valid, interesting or justified. How far we can imagine systems that don’t map to the world in some way might be in question, but no mathematician will ever dismiss a valid mathematical system because it doesn’t map to the real world, or contradicts it. Given that, it is clear that mathematics is not justified by the real world, because nothing in the real world could prove any mathematical system mathematically wrong, false, or invalid.

CoelPost authorHi verbosestoic,

I don’t agree that anything can be known “a priori”, and thus I’m not conceding my critics’ claims. I do agree that pure empiricism alone would not get you anywhere — if you shine data on a rock it won’t learn; you do need a device capable of learning, and that requires some prior programming.

Thus my stance is that we don’t know anything except as a combination of all three of: (1) evolutionary-timescale programming through interaction with the world as a consequence of natural selection, (2) bringing out of that program through our development from an embryo and as a child, and (3) corroboration of the resulting knowledge through direct empirical experience.

I agree that “empiricism” is not perhaps a good word for that combination, but I’m not sure I know of a better one!

Yonatan Fishman‘Knowledge derived from experience’ should be understood to include ‘innate’ information about the world and genetically-guided learning mechanisms acquired via evolution. This evolutionary grounding may account for how we can “know certain things — or at least possess certain concepts — before we can make any sense of experience.”

Yonatan FishmanClearly, a good deal of mathematics (it’s axioms and rules of inference) is abstracted from experience (this explains why it is useful for describing it). However, mathematics can also be regarded as free-floating- independent of experience, insofar as mathematicians are free to make up any axioms they please (like the game of chess). Thus, I think it is helpful to remember that there are really two kinds of mathematics: pure, formal or “game” mathematics, and empirical or “applied” mathematics, with cross-fertilization between the two endeavors. Game mathematics explains the sense of ‘certainty’ that mathematicians experience when they prove theorems, whereas empirical mathematics explains the applicability of mathematics to describing nature. For a chapter length exposition on this view see Simon Altmann’s book, Is Nature Supernatural? (Prometheus books). For a similar view, see the last chapter in Morris Kline’s book, Mathematics and the Physical World. Also recommended is the book by philosopher Carrie Jenkins, reviewed here:

http://ndpr.nd.edu/news/24346-grounding-concepts-an-empirical-basis-for-arithmetical-knowledge/

A relevant article by philosopher John C. Harsanyi:

Mathematics, the Empirical Facts, and Logical Necessity

Author(s): John C. Harsanyi

Erkenntnis (1975-), Vol. 19, No. 1/3, Methodology, Epistemology, and Philosophy of Science (May, 1983), pp. 167-192

John CrothersI agree with you. I would also remind you of the words of the ancient Greek philosopher Parmenides (c.500BC) who said “Our senses deceive us.” We do not observe a space time continuum, we (not me though) merely assume it.

Both mathematics and science do not ask: “How may we develop an abstract theoretical interpretation of the manner in which a real world observer, be it ant, flea, elephant or human (or molecule for that matter – which is merely assumed to have ‘robotic’ behaviour) observes the environment (the universe)?

Why should we humans, who evidently have been made by the universe, decide that we may develop some kind of independent assessment of it? We are clearly observer-centric. This is evident with science, which should be looking to our primary tools of observation, our senses and brain and realizing that we have developed this thing called ‘mathematics’ based on a) real world observations that are completely justified – finite one to one correspondences, and b) assumptions concerning our observations e.g. that we have ‘continuous’ vision or that what we observe is ‘actual’.

We have developed the notion of ‘continuity’ because it an easy notion to take on and theorize about. We have approximately 130 photoreceptors over our eye retinas. We do not notice light falling between them or probably, most of the light that falls on them. We have the illusion that we’re seeing all there is to see. A fly does not see what we do, but do we imagine it sees ‘gaps’ in its vision? How could it? If all of its senses are being satisfied, then this is all that it can understand and observe. It’s completely satisfied. It cannot understand that there are things it is not seeing or sensing. How could it?

The photoreceptors over our eye retinas are arranged probably according to a square Fibonacci lattice with a modulus of approximately 130 million. There are a great many distinct Fibonacci vectors corresponding to a finite simple continued fraction expansion with respect to unit partial quotients, save the first and last: . When we (or any other sighted animal) look at an object’s outline it triggers, via light falling onto the photoreceptors, a set of these Fibonacci vectors and this discrete (and finite) information is passed back to the brain (or intelligent mechanism of the life form). The brain then recognizes this sequence of vectors as a particular ‘geometrical shape’.

There is no evidence that the internal modelling of the environment that is carried out over the senses and brain of an animal relies on some in-built “continuous unit of length” which the mathematician and then the physicist assume.

This universal geometrical arithmetic is beautiful, but of course a lot more complicated than the endless blank page of R^2 or R^n.

Yes, I agree that mathematics ultimately has been derived from our real world observations. The problem is, all sciences and mathematics have become industries which simply will not tolerate questions of a fundamental nature being given an airing.

I’m sure you know what I’m talking about. Anyhow, well done with what you’ve done.

John Crothers

Hans-Richard GrümmYou say in response to a comment:

The whole point is that mathematicians do not spend their time working with mathematical systems built from mathematical axioms that are nonsensical in the real world, and which are directly contrary to the real world.

I would like to submit that the study of p-adic manifolds (or even 4-dimensional exotic R4s) or of Banach spaces withhout a basis are counterexamples to your statement. I agree that much of “classical” mathematics has been developed because it contains excellent models of systems within the physical universe. For the last 100 years, the situation has been different.

Regards,

Hans-Richard Grümm