Ultimate Concepts

     Yes, Gentle Reader, it’s another of those philosophical Mondays, when the news simply doesn’t inspire the spleen and Ye Olde Curmudgeon must reach into his bag of “reserve topics” for an essay. Not that I feel obligated to produce an essay every day, mind you; it’s just that there’s a rhythm to it. It’s a habit that’s easier to maintain than to break.

     In the world of physics, we bandy about terms such as matter and energy and law without troubling to define them too rigorously. We know what they are, you see. We don’t have to engage in Aristotelian hair-splitting to use them effectively. Why waste the time and effort? There’s work to be done!

     But some of those terms have rather interesting implications. Here’s one that I recently spent awhile pondering: What is energy?

     A sophomore physics major has already learned various things about energy, when and where to find it, how to calculate it, and some of the transformations it can undergo. But aside from the situational and effectual aspects, he doesn’t know what it is. It’s not a failing in his knowledge or comprehension; we’re all in the same boat with him. It’s the consequence of the habitual use of an undefined term.

     I’m cheating a bit, here. There are two forms of acceptably rigorous definition:

1. Intensive: This sort of definition possesses a genus and a differentia. It describes the category of entities being defined as a subset of a larger category – the genus — and then states a distinguishing characteristic – the differentia — that entities in the category being defined must possess, but which other entities in the genus do not.
2. Extensive (alternately, Tabulative): A definition of this sort merely lists all the entities that belong to the category being defined. It does not attempt to group them in the ways intensive definitions require.

     If the term energy has a definition, it’s more akin to an extensive one. We recognize it situation-by-situation and law-by-law. For example, in mechanics we recognize kinetic energy, potential energy, and work. In electromagnetics we recognize the energies that arise from interactions between charged particles and magnetic dipoles. In atomic and nuclear physics we recognize binding energy and energy of decay.

     Given the above, the question “What is energy?” divorced from any better specified situation becomes impossible to answer. Though the term is a highly useful placeholder in our analyses and computations, giving it a firm, intensive definition appears to be beyond our capabilities.

     Energy is an example of an ultimate concept: a concept we “know” to be true and useful, but beyond which we cannot see, at least at present.

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     An old gag popular among philosophers runs thus: “Define the universe. Give two examples.” This is plainly an absurd request. Definitions pertain to categories, and the universe, by postulate, is above categorization. It contains everything “real.” But that merely raises another, more fundamental question: “What do we mean by real?”

     Real is yet another ultimate concept. It’s useful, it its way. At any rate we “know” what we mean by it. But it can get rather slippery when invoked in an argument.

     Perceptual variations can cloud the meaning of real. There are many problems in special relativity in which observers in equally valid inertial frames perceive events quite differently. Some of those problems can be quite troubling to him whose working definition of real is “what I can see, hear, smell, taste, or touch that doesn’t go away when I turn my back on it.” The twin paradox is probably the best known, but there are many others.

     Douglas Hofstadter, in his landmark book Godel, Escher, Bach, offered a neat twist on the “define the universe, give two examples” gag. He proposed what we might call the “inverse” question: “Define an entity. Give a counterexample.”

     Real, reality, and entity are all useful placeholders beyond which our human limitations forbid us to see.

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     One more, to epater les mathematical bourgeois. I used this one in On Broken Wings:

     Louis drew three pentacles in one corner of the sheet. "What do you see, Chris?"
     She glanced at him suspiciously. "Three stars."
     He nodded, and drew three rhomboids a little distance away. "And what do you see here?"
     "Three diamonds. What does this have to do with computers?"
     "Patience, Chris. I'm trying to lay some groundwork, here." He drew a large numeral 3 and pointed at it. "And this?"
     "Come on, Louis, get serious."
     He said nothing.
     "It's a three."
     "Three of what?"
     "Huh? Three of anything."
     "Is 'three' a thing, Chris?" He was grinning now.
     "Well...isn't it?" She was beginning to feel confused.
     He shook his head. "Go anywhere you want, in this house or anywhere else, and find me a 'three.' I'll pay big time for it. I've been looking for more than thirty years."
     "All right, what is it, then?" Confusion and frustration were beginning to blend.
     Louis shook his head again. "You're going to tell me. I'll ask a related question." He wrote "Christine" below the 3. "What's this?"
     "It's my name...wait...it's a lot of other people's name, too. It's not me, but it's used to refer to me." She frowned. "Louis, what does this have to do with computers?"
     He declined to acknowledge the question. "What do you call something that's used to refer to something else?" He waited, eyes and grin wide.
     She thought furiously. "A name? A label? A...symbol?"
     His grin blossomed into a brilliant smile. "A symbol. These are both symbols. Nearly pure, too, since they have no use except to refer to other things." He appended "Marie D'Alessandro" to her first name and pointed to it again. "That's a symbol, too. A more specific one, the symbol for you. Now, how does this symbol differ from that symbol there?" He pointed to the numeral again.
     She thought a moment. There had to be a point. She would find it.
     "That," she pointed to her name, "refers to something specific. This," she pointed to the digit, "refers to an idea."
     He laid his pencil down and brought his hands together in three sharp claps. He appeared to be both surprised and pleased.
     "You're on your way, Chris."

     Mathematics, the realm in which propositions can be proved or disproved, constitutes an ultimate. As its rules are unconstrained by “reality,” we can create fanciful situations that have no referent in the world around us. Indeed, high school geometry is exclusively about such situations: two-dimensional situations with absolutely precise mathematical entities in them that follow rigid rules.

     I’m not downplaying the utility of mathematics; I’ll leave that to the “pure” mathematicians, who disdain to have anything to do with concepts that can be applied to practical situations. Indeed, that mathematics of certain varieties can be usefully applied is why it originated...but not why “research” into mathematical systems persists.

     The concept-systems mathematicians explore are ultimates of a unique sort: their explorers don’t expect to find analogues to them in the “real world” (and some of them would be offended if they did). But even the simple stuff, the stuff we all use daily and think we understand completely, possesses an ultimate character. Ponder Louis’s insistence upon three and what it “is” in the above. The “atomic” concept of “three” conceals a foundation that exists only in our minds: cardinality. Only we conceptualizers count. Only we are concerned with enumeration. The objects – hey, another ultimate! – we enumerate are there, self-demonstrating, and indifferent to their “number.”

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     I toy with questions of this kind frequently. The intellectual exercise helps me to keep a good grip on my humility. Given Man’s ever-expanding capabilities, humility is a slippery thing in our hands. But we can recover it by reflecting on the tenuous, contingent, and often disputed ultimate concepts on which all our “knowledge” rests.

     Hoc scio, nihil scio might not have been Socrates’s exact words, but it’s a nice formulation...one we should clasp close at all times. It might help us to avert another Tower of Babel – and you know what became of that.