Category Archives: Mathematics

Complexity – It Ain’t Simple (part 2 of 2)

30 Monday Aug 2021

Posted by in Complexity, Logic, Mathematics, Philosophy of Logic

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The guiding motto in the life of every natural philosopher should be, Seek simplicity and distrust it.”

– Alfred North Whitehead, The Concept of Nature (end of chapter VII.)

Ultimately, the only way we know how to measure the complexity of some process or phenomenon – beyond excruciatingly vague and unhelpful statements like, “this is really complicated” – is by measuring how hard it is to solve the mathematical equations used to characterize the problem. All the rest, even when palpably, indisputably true, is just hand-waving. Sometimes hand-waving makes us feel better, because we need to burn off the energy pent up in our frustration. But it never really tells us anything. On the other hand, we really do have some effective means of measuring how hard it is to solve some mathematical equation or other, and we’ve refined such measures significantly over the past fifty years because such measures tell us a great deal about what we can and cannot do with our beloved computers (which includes all of your portable and handheld devices, in case you weren’t sure.)

Some problems simply cannot be solved. This even despite the fact that the problems in question seem perfectly reasonable ones that are well and clearly formulated. (Actually, being well formulated makes it easier to demonstrate when a problem cannot be solved.) Some problems can be solved, albeit with certain qualifications, while still others are “simply” and demonstrably solvable.i However, saying that a problem is “solvable” – even in the pure and “simple” sense (notice how I keep scare-quoting that word) – doesn’t mean that it can be solved in any useful or practical sense. If the actual computation of a solution ultimately demands more time &/or computer memory space than exists or is possible within the physical universe, then it is unclear how we mere mortals benefit from this theoretical solvability.ii It is these latter considerations that bring us into the realm of computational complexity.

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The Infinite

22 Thursday Jul 2021

Posted by in Logic, Mathematics, Ontology

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I don’t anticipate any explicitly Whiteheadian considerations this time around, but all my thoughts are informed by my Whitehead scholarship, so you never know. What I want to talk about here is the idea of infinity. I say “idea,” rather than “concept,” because even within the relatively constrained bounds of formal mathematics infinity is not one thing. Outside of the bounds of mathematics matters are significantly worse, little or since no effort is made to constrain such talks, or even render it potentially intelligible, with formally legitimate techniques.

Speaking of “outside the bounds,” the ancient Greek word for the infinite is “apeiron” (ἄπειρον), which translates as “unlimited” – the “a” being the negation (“un”) and “peiron” meaning limited or bounded. Clever as they were, the Greeks lacked our additional 2,300 years of mathematical study, so the idea that one can have something that is infinite (unbounded) – for example, the length of the perimeter of a geometrical figure – i.e. a perimeter that exceeds any possible length, measurable either in practice or the ideal, that is nevertheless bounded by an easily measured finite figure (a circle, for example) would never have occurred to them.i But the figure above, the Koch snowflake, is precisely such a figure. (Details can be found HERE. As is my wont, I skip the technical details which will take up more text than this blog post.)

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A = B

10 Monday Jun 2019

Posted by in Logic, Mathematics, measurement, Philosophy of Logic

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Anyone who is reading this post – indeed, anyone who can read at all – has some minimal exposure to mathematical ideas, even if that exposure goes no further than elementary arithmetic and not, as I am only half-jokingly known to say, actual mathematics. (Well, since I’ve mentioned it: the thing I’m known to say is, “that’s not mathematics, that’s arithmetic.” This is always in response to someone who has protested something along the lines, “I’m no good at mathematics; I can’t even balance my checkbook.” The humorous, yet legitimately educative nature of MY statement always strikes me as obvious, yet I am constantly amazed by the numbers of people who get lost in the elementary rhetoric of my statement.)A equals B

In any event, even such minimal exposure is typically enough to satisfy most people, even most mathematicians (I suspect), that they have a pretty good handle on what that equals sign (“=”) means as it is expressed in, say, the title of this little essay. Clearly I wouldn’t be writing about it if such an impression was even remotely true. For one thing, how do we read “A = B”? Does it say, “A equals B”, or does it say “A is B”, and is there a difference between those two? Spoiler: yes. Yes to both questions, depending on how crudely one is using one’s language, which makes the fact that “equals” does not equal “is” an especially problematic conflation of terms. “Is” tends to mean “identity” in such a context, which is tricky enough in its own right (I wrote an MA thesis on the subject). The reading of “=” as “equals” helps to emphasize a somewhat more functional approach to matters, though it is still more rigid and “substantive” than such formal notions as “equivalence” and “isomorphism.” topics I’ll likely blog about in the future because I can already hear the math-phobes screaming in horror. For now, I want to focus on the logical issues of “equals,” as a formal relation. Thus, the word “equality” may also find a use here, but that use should not be mistaken for the political, economic, cultural, &/or social senses of the term. (On the other hand, I do not preclude in advance that what I say here will have no bearing on those uses, either.) Obviously, the starting point for the primary discussion is with the work of Alfred North Whitehead. Continue reading →

Self-Identity

27 Tuesday Feb 2018

Posted by in Logic, Mathematics, Metaphysics, Ontology, Process Philosophy, Relationalism

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I was not an especially “outward looking” or alert youth, working rather to shut the world out rather than invest painful consideration into something that was already almost unbearably painful. But occasionally my habits of thinking would turn themselves outward, to chew on a puzzle that had managed to break through my protective shell and demand my attention. This happened twice that I can recall in high school: the first time, after an especially depressing episode I realized I needed to make a study of reading people – perhaps, more importantly, I realized that I could learn this, and I began picking up clues effectively and rapidly. The second, and first genuinely philosophical moment, was when I “discovered” the “problem of evil” as it related to the born-again Christianity I’d been emotionally bullied into accepting by various members of my family – personal responsibility is a joke, of course, in any world dominated by an omniscient and omnipotent creator god. This began my “angry atheist” phase, which went on for another decade (until I’d actually read a substantial bit of philosophy.)Acropolis1

The third “break through” (second genuinely philosophical one) happened when I was in the army. I was stationed some 18 kliks from what was (at the time) the East German border, in the Central German highlands, as an electronics tech in an Improved HAWK anti-aircraft missile battery. Every year, each such unit chose a squad of people to be sent to NAMFI, Crete, to spend a few days training, culminating in firing a live bird at a drone target. As it happened (then, at least), the entire trip involved several days both before and after the actual training which were free time for the troops to explore the island or, as several of us chose to do, take the ferry from Souda bay to the Piraeus and Athens. So it came to pass that I climbed the steps up the hill of the Acropolis. Except, that’s not quite right. Nobody actually walked on those steps, and it wasn’t out of respect for their antiquity. Continue reading →

Intuition in Mathematics and Physics: A Whiteheadian Approach

05 Friday Aug 2016

Posted by in 2015 International Whitehead Conference, General Philosophy, Inquiry, Logic, Mathematics, naturalism, Philosophy of Science, Public Philosophy, Whitehead

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This is just a quick shout-out to my friend and colleague, Ronny Desmet, for putting together the papers that were presented at the 2015 International Whitehead conference in the new book, Intuition in Mathematics and Physics: A Whiteheadian Approach, in which yours truly is a contributor. Intuition Mathematics

The articles within are from Section IV, Track 2 of the conference. The table of contents is not yet available at Amazon, so the contributions are as follows:

  1. Integral Philosophy – An Essay on Speculative Philosophy – Ronald Preston Phipps
  2. Reflection on Intuition, Physics, and Speculative Philosophy – Timothy E. Eastman
  3. Whitehead on Intuition – Implications for Science and Civilization – Farzad Mahootian
  4. Whitehead’s Notion of Intuitive Recognition – Ronny Desmet
  5. The Beauty of the Two-Color Sphere Problem – Ronny Desmet
  6. The Complementary Faces of Mathematical Beauty – Jean Paul van Bendegam and Ronny Desmet
  7. Creating a New Mathematics – Aran Gare
  8. Whitehead, Intuition, and Radical Empiricism – Gary Herstein
  9. What Does a Particle Know? Information and Entaglement – Robert J. Valenza
  10. A Neurobiological Basis of Intuition – Jesse Bettinger

Rhythm and … Logic?

30 Monday Nov 2015

Posted by in Aesthetics, Critical Thinking, Logic, Mathematics, Whitehead

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There’s a false dichotomy which supposedly stands between aesthetics and analysis. But art and emotion do not stand in opposition to logic and reason. This nonsense is, in many ways, the bastard offspring of the “two cultures” story we’ve inherited since before C. P. Snow gave it a name, and which we’ve variously integrated into our teaching programs for almost all levels of education. Back in the “good old days” of classical education (by which I mean the ancient Greeks) mathematics and music were treated as much the same thing. Even today, we have not quite lost all sight of those connections, and if one takes the time to listen to mathematicians, one will notice that the issues of whether a proof or a theorem is beautiful or not takes on primary importance.DrumStickNylonPic

http://www.youtube.com/watch?v=1XWlK-cL504

Careful, meticulous reasoning is not cold; quite the contrary, it is a fire that will consume you without mercy. I’ve touched on the idea of mathematics and the beautiful before, but wish to revisit the idea again because it can bear the company, even in this Thanksgiving season. This time around, however, I wish to approach matters from a more “musical” perspective that specifically highlights some ideas around “rhythm.” I mean to tackle these ideas from what I take to be a very Whiteheadian point of view. Whitehead was, of course, an accomplished mathematician and educator, and well attuned to the subtleties of mathematical aesthetics. But as he began to worry about the philosophical underpinnings of our physical sciences, his inquiries began to lead him from issues of organization (of thought) to organism itself. Rhythm became one of Whitehead’s central concepts. Continue reading →