What is the relationship between mathematics and science?
Not necessary, not close - as any perusal of the major breakthroughs in the history of science will reveal.
But people who are good at mathematics, and who don't know much about the full range of science, like to assert that all "real" science is mathematical, or that maths is the key to science, or that maths expertise means you are a kind of arbiter of science...
Maths is one thing; science is many other things - some of which overlap with maths.
In a nutshell, science makes models of bits of reality. Some models concern reality is structured - these are qualitative models, usually made of categories - and are not mathematical - or only seldom. Some models concern how the world functions, and some of these are mathematical.
Maths is also a kind of model, but not a model of the world but a model of mathematics, Some bits of this mathematical model can be used to represent bits of science: i.e. some bits of maths can become scientific models (which represent bits of reality).
Unless you are a special kind of Pythagorean or Platonist; here is no a priori reason why mathematical operations should necessarily be the same as all the functional operations of real life - and they aren't.
We do not live in a reality underpinned by the abstraction that is mathematics - but a reality underpinned by God, who is personal.
So; both ultimately and in practice, the relationship between mathematics and real science is variable and not close; although maths (numbers and calculations, statistics and mathematical models) has and is often been the basis of the varieties of pseudo-science - an activity designed to manipulate people, rather than to understand the work.
3 comments:
"So; both ultimately and in practice, the relationship between mathematics and real science is variable and not close "
I agree.
I don't believe that knowledge of mathematics is essential for the sciences or even that everyone needs to know mathematics. While it has often been part of a general education over the centuries, the special study of mathematics has pretty much always been a minority interest. Though unlike physicists or computer programmers trying to tell everybody else how to think (in the 20th and 21st centuries respectively), mathematics has never really done that. It's been off in its own niche, pursuing its own problems, though naturally those often arise from other fields.
I do believe mathematics is part of reality, but how exactly it relates to the rest of it, I'm not sure. One of the reasons I believe it is real is its depth. It's not infrequent that one encounters a movie, a video game, an imagined world of some kind, where the world doesn't feel real, parts of it feel arbitrary. Though depth is not dependent on technical craftsmanship or complexity. Tolkien's Middle Earth is a famous example of an imagined world with depth. There's a famous technology that has supposedly limitless potential, but needs to vaccuum up vast quantities of data to produce anything. That's the opposite of depth.
But mathematics has a great deal of depth. It's true that people can think about various political ideologies, such as Marxism, for decades, but mathematics has something more than that. There are frequent surprises and new unexpected ideas that come about just from investigating mathematical objects. Rather than exhausting and coming to the end of the investigation, more and more things keep being produced. (Though It's necessary to have made a focused study of mathematics to experience this for oneself).
So, I agree that these ideas: "all "real" science is mathematical, or that maths is the key to science, or that maths expertise means you are a kind of arbiter of science " are mistaken. But as its own discipline (which no one is obligated to pursue) I would say that mathematics is both real (though it's difficult to say exactly how) and overall good.
@NLR - I know what you mean, not from experience but from reading the biographies of creative mathematicians.
But the same is true of sciences, if you are among the few who choose to explore them in this way (I was one of this few). One difference is that the sciences change their paradigms from time to time, and this cuts off "depth" from the time before the paradigm change.
For example, biology changed its paradigm in the 1950-60s, from being the study of living things, to the study of entities subject to natural selection - changing focus from life to replication, in effect.
But mathematics has built upon a paradigm going back for more than 2000 years - and so is much more extensive - much deeper and more interlinked.
This happens - probably - because science is more linked with the rest of the world than is mathematics - which is essentially autonomous.
Another factor is that scientists are often very orientated towards practice, observation, "empiricism" - even to the point of active hostility to the kind of theoretical exploration of the field (testing for internal consistency, and implications etc) that is the business of pure mathematics. The knock on effect is that pure mathematicians have colleagues (and the advantage of communicating in a common language); while a theoretical biologist may have literally nobody to talk with in his area of interest.
Having genuine colleagues is usually very helpful in accelerating progress, and establishing a basis of solid (tested) knowledge - and conversely a lack of colleagues (and lack of interest in findings) has the opposite effect.
Interesting discussion. I was formally trained math and physics, then moved into microbiology, and engineering. Math as a discipline is wide, varied, and ultimately theoretical. In the end, that portion of math that I found was useful to "real life" was statistics. By observing whatever you are interested in and quantifying what you observe, you can gain some insight into that whatever. This helps you to understand and resolve or problem solve. I am not sure that topology or even Lagrange multipliers contribute much to real world understanding. Then, ultimately, I agree that the only really important pursuit is God.
As I was ready to press the publish button, it does occur to me that in-depth math study immerses one into a world with great beauty and elegance. This is much like music. Yes, I was a semi-professional musician for a short time, but I discovered I liked to eat more. :-) But it does strike me that math is more of a right brain activity like music or art where engineering is more of left brain activity. I need to think about this more. hmmmmm.
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