I have to admit to pretty much agreeing with Coel’s main points in this post, which anyone who read my recent post on logic probably won’t find too surprising. The idea of math and logic resting on empirical foundations seems to be ferociously resisted, I think because those foundations don’t feel empirical, mainly because we don’t learn them empirically. The human brain is not a blank slate. It comes with pre-wiring for a number of capacities, including logic and some math. We don’t always use it, but we evolved it, probably due to its survival advantages.
However, unlike Coel, I’m not insistent on mathematics being a part of science. I’m content to leave science to endeavors that involve a heavy amount of empirical investigation, and the logical and mathematical consequences of that investigation. Mathematics may have empirical foundations, but I think it’s pretty obvious that mathematicians aren’t doing empirical work, but finding interesting and (sometimes) useful tautologies.
[Editor’s Note: This essay is part of Scientia Salon’s special “scientism week” and could profitably be read alongside other entries on the same topic on this site, such as this one by John Shook and this one by yours truly. My take on the issue is very different from that of the authors who contributed to this special series, and indeed close to that of Putnam and Popper — as it should be clear from a recent presentation I did at a workshop on scientism I organized. Also, contra the author of the third essay in this series (but, interestingly, not the author of the first two!) I think the notion that mathematics is a part of science is fundamentally indefensible. Then again, part of the point of the SciSal project is to offer a forum for a variety of thoughtful perspectives, not just to serve as an echo chamber…
View original post 3,356 more words
SelfAwarePatterns 
Just curious. What would you say about a line defined as a breadth-less length? You can’t “see” this line; it’s impossible. How could a breadth-less length be empirically founded?
Or am I being really passé in even mentioning Euclid? Is that definition still accepted?
I’m not being facetious here. I really have no comprehension of this non-Euclidian Einstein stuff.
LikeLike
For me, and I may be somewhat different than Coel on this, mathematics has foundations that are empirically observable (even if we don’t necessarily learn them empirically) but then is logically extrapolated far beyond those foundations. Most of what actually happens in mathematics, other than the simplest operations like counting and very basic arithmetic, are those extrapolations, the formation of tautologies.
So, my thinking is that a breadth-less line would be one of those extrapolations, a useful logical construct. But it is ultimately derived from relationships that exist in the world, otherwise we wouldn’t find it useful.
Hope that’s clear.
LikeLike
I think I understand, but not sure.
So then some mathematics such as simple operations, 1+1=2, for instance, are not empirically derived, but others, such as a breadth-less length, are extrapolated from experiences of visible lines?
LikeLike
The relationship of mathematical objects to entities in the world are often not easily traceable (hence the ongoing debate). For a breadth-less line, my guess is that it would come from multiple observable entities in the world: observations of visible lines, and extrapolations of the mathematical relationships that manifest as lines.
I guess the reason I think that such a relationship exists, despite my inability to definitively nail it down, is that I can’t think of a better alternate explanation for why these entities are useful in the real world.
LikeLike
Thanks for the thoughtful reply. I had read that blog post earlier and didn’t want to comment there as discourse seemed to be getting a little nasty.
I’m using this example in my novel to discuss the ideal vs. real. So this discussion is very helpful for me, especially since my understanding of math boils down to calculating tips at restaurants.
I suppose it can go both ways: Do I see lines in the real world because I have this breadth-less length idealization of a line or do I have my idealization as an extrapolation from real world lines?
But the funny thing is, the experience of the ideal line and the visible one come at the same time, so you’re right, it’s hard to nail down. It’s hard to say which came first, or if one is indeed prior to another.
The whole rationalist/empiricist debate gets confusing to me because so much of what we’re talking about seems to be two sides to the same coin. Ideas are within experience too, even though they aren’t visible entities. I feel that if I’m being true to experience, I can’t just dismiss them as concoctions or dream clouds the way William James does in his radical empiricism. He even goes so far as to try to describe experience without ideas as being some kind of drunken state, or dizziness. I don’t think we can have experience at all without ideas, and it doesn’t seem useful to try to get at little nuggets of “pure” experience in the manner of WJ.
That said, we still don’t know where ideas come from or if they’re dependent on the ‘real’ world, or visible entities. All I know is that I need that line drawn on a sheet of paper to even think about the ideal line, even though they are quite distinct.
Interesting topic, by the way. Thanks for posting this and replying so thoughtfully!
LikeLiked by 1 person
rung2diotimasladder, perhaps you can think of the abstraction “line” in the same way as the abstraction “chair”. Both are derived from our real-world observation of “things” and are then subjected to our “thinking” process.
Treating every chair as a unique object would be hard work and would stop us seeing the bigger picture. There are survival advantages in abstraction, as well as aesthetic ones.
I’m generally comfortable with the idea of Mathematics as an empirically-rooted science (how would we ever have invented numbers if we didn’t have sheep to count?), but where does Logic come from? Are logical rules “out there” in a Platonic sense? Or are they more akin to human-devised preferences based on ideas of order and beauty?
LikeLike
Steve, I think the empirical foundations of logic are harder to visualize, but that the relationship is similar. How do we know whether a logical construct is valid? We may refer to other logic to evaluate it, but how do we know that logic is valid? Eventually, don’t we have to refer to how things work in the world?
If logic is “out there” completely divorced from reality, then why is it useful? To me, this is just the other side of the coin of the “unreasonable usefulness of mathematics”.
LikeLike
A younger version of myself would have argued the Platonic case, but I have moved 180 degrees and now tend towards the same empirical vision that you describe here.
On most big scientific and philosophical questions, my stance is usually “we just don’t know yet” but I believe that one day we will. Otherwise, we live in a senseless universe.
LikeLike
Your younger self sounds more thoughtful than my younger self. Sometimes, when I ponder how different my positions today are from myself 20 years ago, it makes me wonder how different my positions might be in 10 or 20 years. Will I find that I’m currently close to truth with only fine tuning required, or are there dramatic paradigm shifts to come?
On the big questions, I think that’s an accurate assessment. All we can do is speculate and reason what we think might be the answer, but we have no way to insure our speculations aren’t hopelessly biased or blinkered.
LikeLike