The unity of maths and physics revisited

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:

The roots of empiricism: Hume’s fork, and the divide between knowledge “by observation” and “by reason”

Scientia Salon recently published my article advocating that mathematics is best regarded as a part of science. In reply to “scientism week”, Massimo Pigliucci wrote an article criticising “the return of radical empiricism”. The collision of “scientism week” with “anti-scientism week” generated a lot of energy and comments!

Massimo Pigliucci’s article is well worth reading, being a clear exposition of the relevant ideas. He traces the issues back to Hume’s famous fork, in which Hume declares that:

All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact and real existence.

The “relations of ideas” category is taken to include mathematics and logic, where knowledge is “discoverable by the mere operation of thought”, while the “matters of fact” category contains science, where knowledge derives from empirical data.

Kant rejected Hume’s empiricism and sought to establish the primacy of reason. He adopted the term “a priori” for knowledge that does not derive from experience, in contrast to “a posteriori” knowledge which does. A related concept is that of “analytic” statements, which follow from the definitions of the terms, contrasting with “synthetic” statements that describe how the world is.

This notion of a fundamental epistemological divide holds today, and is at the heart of resistance to the idea that mathematics, logic and science are a unified whole.

In reading Pigliucci’s article I agree with much of what he says, but, to me, he seems to miss the main arguments for the essential unity of the different domains of knowledge. I will thus outline how I see the roots of empiricism, and then consider the supposed divide between knowledge “from reasoning” versus knowledge “from observation”.

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.

What does “existence” mean?

During a recent online discussion I discovered, somewhat to my surprise, that there is no general agreement on what the word “exist” means. Everyone has an intuitive understanding of it but when it comes to an explicit definition of the word there is no consensus, and indeed philosophers have written a vast literature on the topic of ontology, or what exists.

Dictionaries don’t really help; for example Oxford Dictionaries gives a nicely circular set of definitions:

Exist: 1. have objective reality or being
Reality: 1. the state of things as they actually exist
Being: 1. existence.

Of course physicists have a perfectly good operational definition: something exists if it is capable of making a detector go ping. Try arguing that, however, and you’re immediately accused of materialism, physicalism, scientism, being blind to possibilities beyond a very narrow world-view, and a host of similar sins (I plead guilty to at least the first three).

Disagreeing (partially) with Sean Carroll about what is science

Sean Carroll talks a lot of sense on the nature of science, and in a recent post gave a definition of “science” that focuses on the methods and attitudes of science. He wrote:

Science consists of the following three-part process:
1. Think of every possible way the world could be. Label each way an “hypothesis”.
2. Look at how the world actually is. Call what you see “data” (or “evidence”).
3. Where possible, choose the hypothesis that provides the best fit to the data.

It’s a good operational definition, separating a scientific attitude from a pseudo-scientific or religious attitude, although the process needs to be iterative, with the hypothesis of step 3 then being tested against new data; and I would add in Feynman’s maxim: try hard to see whether you are fooling yourself, remembering that yourself is the easiest person to fool.

Later in the article, however, I start disagreeing with Carroll about what is and isn’t science. He says:

Scientism and questions science cannot answer

“Scientism” is often taken as the claim that science can answer all questions. Of course there are plenty of things that scientists don’t currently know, so the suggestion is, instead, that science could potentially answer all questions, or at least all meaningful questions.

For example the philosopher Julian Baggini says that

“What is disparagingly called scientism insists that, if a question isn’t amenable to scientific solution, it is not a serious question at all.”

Another noted philosopher, Massimo Pigliucci, writes in his book Nonsense on Stilts :

“The term “scientism” encapsulates the intellectual arrogance of some scientists who think that, given enough time and especially financial resources, science will be able to answer whatever meaningful question we may wish to pose …”

I disagree with these definitions (both of course by people critical of scientism), and suggest that scientism is instead the claim that science can answer all questions to which we can know the answer. The point is that there are many questions that are “meaningful”, yet we can never, even in principle, answer them. First let’s distinguish between meaningful and meaningless questions.