Is logic and mathematics part of science?

Last week was scientism week at Scientia Salon, and I reblogged a post by Coel Hellier on a defense of scientism, mostly by arguing that mathematics was actually part of science.  As I indicated in my comment on that reblog, while I agree with Coel that both logic and mathematics have foundations that are empirically observable (my wording here is actually pretty careful; for why, see this post), I’m not really on board with calling mathematics a science.

To some degree, I see this as a bit of semantic issue.  Whether or not to include certain fields as a science ultimately comes down to what your definition of science is.  Some have a fairly narrow view of science, others a broader one.  Massimo Pigliucci, in the latest Rationally Speaking podcast, stated that he sees science as those subjects that have historically been labeled as science.  I can’t agree with that view, mainly because I can’t see how any new scientific fields could ever be started with it.  My view of science is fairly expansive.

To me, science is the pursuit of reliable knowledge about reality.  Part of that reliable knowledge are the methods of investigation that have been shown to be reliable in acquiring that knowledge, in other words, the scientific methods.  “Reliable” in this context means having a low probability of being substantially revised in the future.  (“Substantially” because just about every scientific idea eventually gets revised in at least small ways.)  Note that there’s no bright line between reliable and unreliable, just a continuum, with science usually concerned with the most reliable things that can be demonstrated.

I have to admit that this definition is a slightly broader one than I’ve written about before.  My earlier definition was pursuit of reliable knowledge on how reality works.  The extra word restricts science to discovering natural laws, but I think it omits things like figuring out what actually happened in the early universe, or the geological history of Earth.

But taking this slightly broader view means that it ends up including fields like history and journalism, as well as things like good police detective work and plumbing.  I personally don’t have a problem with this.  It explains what separates, say, rigorous journalism from blatant opinion, but many people will regard it as a form of scientism.

But I think this definition excludes pure logic and mathematics.  The reason is that while these fields can produce reliable knowledge, it isn’t necessarily knowledge about reality.  Examples include abstract concepts like the Mandelbrot set, which apparently have no real world correlate.  Of course, this can also result in an argument about the definition of “reality”, a rabbit hole that I don’t think I’ll jump down today.

But wait a minute.  If logic and mathematics do indeed have foundations that are empirically observable, then how can logically correct extrapolations from these foundations produce something that doesn’t match reality?  How can we get the Mandelbrot set from real world relations if the result isn’t in the real world?  I’m not going to pretend like I understand abstract mathematics, but based on discussions I’ve read from people who do, I think the answer is that we get them because the foundations, while grounded in reality, aren’t a complete account of reality.

Consider if I decided to build a model of the universe, but only using the laws of electromagnetism.  That is, I ignored general relativity and any other scientific understandings.  My model would be grounded in reality, but in an incomplete account of that reality.  The resulting model would not represent much of what is actually in the universe.

Of course, no scientist would do this, at least not on purpose.  (They obviously do it accidentally all the time while constructing speculative theories).  The reason they wouldn’t set out to do it is, it is unlikely to tell them anything interesting about reality.  But mathematicians and logicians do it all the time.  And in the case of mathematics and logic, it does often produce interesting constructs, that occasionally turn out to have real world correlates.

Now, if your definition of science is pursuit of reliable knowledge, period, then I can see a case being made for including mathematics within the science umbrella.  However, while my view of science is expansive, removing reality from the definition strikes me as ignoring what motivates most of science, which is to learn as much about the world as possible.  It seems like an important distinction to me.

That said, if you look at how departments are organized in academia, the math department is sometimes lumped together with the science departments and sometimes separated, so there’s obviously people in academia who fall into both camps.  Although admittedly organizational structure often has more to do with politics than with particular philosophies about what is and isn’t science.  Still, putting mathematics in the science division or college is a feasible move because many people believe it belongs there.

Logic on the other hand, is generally in the philosophy department, which in modern organizations is almost never grouped with science.  This is probably so because formal logic is rarely used in science, if ever.

So, while I personally don’t think logic or mathematics is science, despite the incredible usefulness of mathematics in science, I’m also not adamantly against the idea.