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I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce.

Freeman Dyson

Is there a philosopher who articulates the difference between how a modern physicist and mathematician approach science? (Not sociological differences)

This becomes apparently clear when we get into interdisciplinary discussions. I think it's more than rigour. Rather it seems to me they are thinking about reality differently? A source that suggests so (adding more):

https://mathoverflow.net/a/80531/64357

https://mathoverflow.net/questions/116531/the-unreasonable-effectiveness-of-physics-in-mathematics-why-what-how-to-catc

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Sweeping generalisation alert. Physicists tend to be very pragmatic. If they can find a mathematical technique that predicts the results of experiments, they're happy- they won't have sleepless nights worrying about whether its validity is utterly watertight in a mathematical sense.

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  • This is the correct answer. An example would be renormalization. When renormalization was first introduced its mathematical foundation was very unstable.
    – causative
    Commented Sep 4, 2023 at 16:54
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Mathematicians need not practice science at all, except as a personal hobby unrelated to their profession. If you search Physics SE for the inverse of this question - "how does the physicist's approach to mathematics differ from the mathematician's?" - you'll find useful results, such as: here and here. I recommend Anna V's Feb. 2017 response at the first link.

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Alright, this question begs for some clarification since it seems to equivocate a little on 'science'.

The demarcation of science, as Karl Popper called it, is a question about determining what is and isn't science. While we often hear children taught about 'the scientific method', any sustained inquiry suggests that there is at best a generalized abstraction that we use to suggest some of the more salient features of scientific activity, such as falsifiability as discussed by Popper at length.

But, science is used also in the sense of 'reliable knowledge' long before the modern 'scientific method' as might be understood in the 21st century. We therefore must note that one often sees several types of sciences:

  • the formal sciences (math and logic)
  • the natural or physical sciences (sciences of physics, biology, and chem, geology, etc.)
  • the social sciences (sciences of mind and society)

Difference between how a physicist and mathematician approach science?

So, if you mean by this question, and it is not clear, how does each approach their own discipline differently, then a physicist practices physical science and a mathematician practices formal sciences, and a mathematical physicist is somewhere in between. A physicist relies on experimentation primarily (think Galileo Galilei dropping objects from the Torre at Pisa and using inductive logic) and a mathematician relies on calculations (think Euclid's Elements and lots of deduction). While strictly speaking, human reason and its drive to explain goes far beyond both induction and deduction, there's a general difference of what might be described as classical rationalism and classical empiricism. In effect, a mathematician uses thought as the source of his knowledge, and a scientist uses observation. This of course is a massive oversimplification, as it turns out, that in the philosophy of science it has become accepted that observations are theory laden, and that there may or may not by a Myth of the Given (SEP) depending whether you believe Sellars or McDowell. Simply put, our thoughts affect our observations, and our observations affect our thoughts, at least if you are willing to accept mental causation (SEP).

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  • "think Galileo Galilei dropping objects from the Torre at Pisa and using inductive logic" There is no clear evidence about Galileo having really performed the experiment. There is no "inductive logic" (whatever it may be) in Galileo's work.
    – Mauro ALLEGRANZA
    Commented Sep 5, 2023 at 7:59
  • @MauroALLEGRANZA Nor evidence Netwon was hit in the head with an apple. But apocrypha are often instructive and facilitate communication as figurative idiom. As such, the facticity in my passage is irrelevant, and your comment will serve as a disclaimer for the unwary. As for "inductive logic", don't be so literal. The mathematization of science necessarily entails underdetermination of hypothesis, and as such, clearly cannot be deductive in the greater explanatory framework...
    – J D
    Commented Sep 5, 2023 at 16:16
  • The leap from measurements to conclusion is enumerative or eliminative induction. Don't confuse the map for the territory. Just because the language had been constructed to describe the species of logic doesn't mean that the logic wasn't of the species.
    – J D
    Commented Sep 5, 2023 at 16:16
  • Newton's hypothesis about gravitation was imagined and then tested with a lot of computations, comparing the results with tables of astronomical observations and corrected taking into account necessary simplifications (two bodies problem). No induction at all. The same for Galileo's conjecture about the law of free fall.
    – Mauro ALLEGRANZA
    Commented Sep 5, 2023 at 16:50
  • It is not from measurement to conclusion, but from conjecture to prediction to computing effect to be tested and comparison with measurement, correcting for discrepanze.
    – Mauro ALLEGRANZA
    Commented Sep 5, 2023 at 16:52
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Is a theoretical Physicist a physicist? Is an applied mathematician a mathematician?

If so, then I believe there isn't a formal distinction in how either approach problems.

Now, there are general, tendancies towards differences.

Phycisists have a grounding to them, they care about contact with reality.

This effects methods and practices, as they restrict themsleves to things that they believe represent the world.

Mathematicians have no such grounding, they are free to engage in enterprises which on their face, seem to never have contact with reality, it's not that they think about reality differently than a physicist does, it's that they don't think about "reality" ( unless they are a Platonist!!!) at all- in their role as a Mathematician.

The differences between a platonist and a physicist are quite a bit harder to narrow down.

If pressed for a simple demarcation, I'd describe the distinction as Phycisists study the laws of Nature, and their effects, while Mathematicians study the laws of logic, and their effects.

The degree to which the laws of logic, and nature are actually distinct- I believe is an open question.

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    I think it's more accurate to say physicists and mathematicians have different kinds of grounding. Mathematics is grounded in the laws of logic; physicists can sometimes afford to make leaps that are not justified by logic alone because their discipline is grounded in physical reality.
    – Charles Staats
    Commented Sep 5, 2023 at 14:18
  • That's a fair point. I do want to add that the nature of the relationship to the grounding is different. Phycists can't change the laws of nature, mathematicians are free to work under their own newly invented laws of logic. This may not always be the case, maybe the Physicists of the future will have a more malleable relationship to fundamental natural law. A physicists grounding to nature seems to be bedrock, a mathematicians grounding in logic, is floating on a cloud.
    – Michael Carey
    Commented Sep 5, 2023 at 15:47
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    Mathematical laws of logic are grounded by existing work: Any new proposed system of logic that was shown to contradict even a recently-proven theorem (without pointing out a flaw in the proof that would already have been considered problematic) would be treated with extreme skepticism. There are mathematicians who study nonstandard systems like intuitionism, but it's a pretty niche area precisely because these systems are too weak to support the existing body of mathematical work.
    – Charles Staats
    Commented Sep 5, 2023 at 20:46
  • That's all very true!
    – Michael Carey
    Commented Sep 5, 2023 at 21:40
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It is similar to the difference between a designer and a mechanical engineer working on a merry-go-round. The designer (or physicist) is concerned ultimately with some observable phenomena and general behavior of the entire system. The mechanical engineer (or mathematician) is concerned with gaining complete understanding of the components, their precise limitations, and how they work together.

The mechanical engineer may spend a long time studying the threading of one of the bolts. After that study, they are incredibly confident of how it will behave in certain circumstances. Analysis of all such components being used in the merry-go-round could then be used in principle to rigorously analyze the behavior of the entire system, although it may be computationally infeasible.

The designer, on the other hand, may be asking more broad-scoped questions like, "Will the kids riding it be flung off at this speed?". You can answer such questions without modeling the bolt threading, maybe just simplify the whole system to a spinning disk and do a rough calculation. Maybe later they wonder if the animals need to be attached at both the bottom and top, or if they could be supported just from below. They may then move to a more refined model with spherical animals, but again not on the level of needing to know about the threading of the bolts. Part of their expertise is in making unfaithful simplifications of the problem without changing the answer too much.

What's important to stress here is that actually neither of these activities are necessarily scientific in themselves. Such analyses can be done even if there don't exist any merry-go-rounds at all. Many people do not consider mathematics to be a science at all (although it may be used as part of a scientific theory). But even more unexpectedly much of theoretical physics also is not science. For example, there was an immense effort to describe orbits of planets (including "retrograde motion") using nested epicycles. It turns out that by general mathematical principles (later formalized by Fourier) any motion observed could have been modeled using epicycles. So there is actually zero scientific content in the statement "planets move according to nested epicycles": it does not make any testable (falsifiable) predictions about the motions of the planets. If the prediction with two nested epicycles is wrong, you can add a third, or a fourth, or a fifth, etc. to accommodate any empirical challenge to that claim.

There is somewhat of an epistemic difference, though. A physicist might gain confidence in a certain novel inferential technique by creating predictions that rely upon it, and experimentally validating those predictions. This allows them to combine deductive and inductive knowledge in a somewhat unique way (compared to, say, an epidemiologist). What is unique about it is that physicists will gain confidence in inferences that are very high level (nested deep in the theory itself, not directly connected to the phenomenon being observed). Think for example about how separated CPT (Charge-Parity-Time) symmetry is, as a principle, from specific predictions about a particular physical system. This makes it difficult to say what the ontology of a physical theory is: are the epicycles "real" / "physical", or just theoretical artifact? Is the "wavefunction" real or just a mathematical convenience? Is the universe "really just a hologram"?

It is probably more accurate to view theoretical physics as a non-scientific quasi-mathematical discipline which bases its rules of inference on a combination of analytical knowledge and empirical knowledge, rather than as a scientific field in itself. After all, most of the "models" considered in theoretical physics are not even intended to model the real world.

In mathematics, we typically permit only analytical rules of inference, and again it is not really scientific because it makes no predictions. The ontology of mathematics incredibly liberal, making even the wildest of mereologists blush. The way mathematics is is used in science is more similar to the way English is used in science: it is used to express a scientific theory, but is not science in itself.

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Looking at Dyson's quote in context (from the talk "Missed opportunities", Bull. Amer. Math. Soc., 1972), it appears he thought that the divorce had happened as early as the 1860s. One of his examples of a missed opportunity was mathematicians' failure to study Maxwell's equations in depth—which likely would have led them to discover special relativity before Einstein—because "mathematics in the later nineteenth century had developed in quite other directions" (among other reasons). He contrasts that with mathematicians' reaction to Newtonian mechanics 200 years earlier.

It's true that mathematicians and physicists "see the world differently" in a way that's well articulated by the answer of Jeff Harvey's that you linked, but I think that's unrelated to what Dyson was talking about. On the contrary, Dyson wanted mathematicians and physicists to collaborate because those different approaches complement each other. What Dyson was talking about was mathematicians' increasing lack of interest in studying theories that have a discernible connection to physical reality—in contrast to past centuries when mathematics was very much about understanding the physical world. I think that is essentially sociological/cultural.

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I wonder if: Physics studies what is observable. Mathematics studies what is establishable *) **).

*) Maybe a subset of all that is establishable, but it does not study anything that is directly observable.

**) .. yet more or less linkable to what is observable, sometimes in very strange ways.

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