Zermelo was the first to axiomatize Set Theory, just one hundred years ago (1908). It was a pioneering, rough-and-ready approach, later refined by Fraenkel; the resulting system, now a standard, is denoted
ZF.
 |
| Herr Dr. Fraenkel |
 |
| Herr Dr. Zermelo |
Another tool in the set-theorist’s kit, often resorted to but to be used only with caution, is the
Axiom of Choice, which has (quite surprisingly) been proved independent of the other axioms. (That is to say: starting from ZF, one can conclude neither to the truth of the Axiom of Choice, nor to its falsity.) When you add Choice as a further axiom, all sorts of amazing and at times disturbingly paradoxical things can now be developed; most notoriously, the Banach-Tarski paradox, whereby you can slice an apple into a
finite number of ingeniously gerrymandered pieces, then reassemble these into another apple, of the same shape but twice the size (with no gaps) (By "can" of course I mean 'could if you were an angel who could manipulate the continuum into non-measurable sets'. So, don't try this in your kitchen.) This is Set Theory in the sort of red-blooded miraculous mode that would have thrilled
Chesterton; he would doubtless have been reminded of the miracle of the loaves and the fishes.
The axiom system that includes this dangerous addition is known as ZFC.
(Normally an essay just rolls merrily along; but let us pause here. If you have really taken in everything in the preceding paragraph, you probably need to lie down.)
* * *
~ Commercial break ~
Relief for
beleaguered Nook lovers!
We now return you to
your regularly scheduled essay.
* * *
We may posit something similar for Reality, by which I mean the (hypothesized) seamless oneness of the observable physical world and our experienced noösphere. (Of course, in practice, we know as little of this as a starfish knows of the stars. But play along -- this is a thought-experiment.)
Z, let us say (we are launching off on a wild metaphor; hold on to your hats), now stands for
Zentral, and axiomatizes the empirical core – the laws of physics.
F (for
Furthermore) adds the necessary regimentation, of these and much else – the laws of mathematics. Materialists, autists, and logical atomists stop right there; a dismal landscape.
But C – ah, C! We cannot be true to the life we lead without adding that. And once again it stands for Choice, but now in the sense in which we daily know it: free will.
(Yes it’s a stretch, I cannot defend it; but it is more than a mere pun, it is a bridge between the core of our being, and the core of math.)
We are actually in better epistemological shape here with this new human-centered ZFC, than we were with Set Theory. In neither the one nor the other can we derive Choice from the other axioms; but by their fruits ye shall know them. (Cf. Gödel: “There exists another (though only probable) criterion of the truth of mathematical axioms, namely their fruitfulness.”) We have never beheld a Banach-Tarski partition-and-reassembly, and are unsettled by the very idea; whereas we experience our free will constantly. And as in the case of Set Theory, once you have the whole ZFC to work with, a heck of a lot of things may logically follow. Gödel himself worked on an aspect of this problem: a formal derivation of (some refinement of) Anselm’s Ontological Argument. And this, from a stance of sheer logic, not a credo, let alone credulity. He came, in fact, to doubt set-theoretic Choice – though the key point here is that, as a Realist, he was sure there was a Fact of the Matter, despite that axiom’s logical independence, an independence which he himself had earlier contributed to proving.
Gödel might in this area be dismissed as an eccentric, or a raised-from-the-dead Leibnizian, his speculations the phantom fruits of anorexia; but we encounter a similar post-Scholastic optimism in a man who liked his beef:
“Morality is capable of demonstration, as well as mathematics.” (Locke, Essay, III.xi.16)
Locke does not go on to argue or develop this idea; I cite the apophthegm more by way of celebrity endorsement.
*
A word in about the proper place of axioms (while noting their similarity to tenets of the Creed).
Confronted with variant axiomatizations of Set Theory (besides ZF, there’s von-Neumann-Bernays, and a number of others), of which several may be used or considered by the same researchers, and none anathemetized, one could get the misimpression that it’s all a game. “What shall we do today, gang? -- Let’s toss together some axioms and put on a show!” Certainly most religions take themselves with more exclusivity (and thus, perhaps, with more apparent seriousness); one is not a Muslim on Monday, and a Hindu on Tuesday. But in fact the whole set-theoretic enterprise began as an essentially empirical investigation of a perceived (though invisible) reality, the World of Sets. The axioms came later, as a counterpunch to paradox and mounting complexity. Thus, van Heijenoort, on Thoralf Skolem’s very technical early work:
Skolem .. does not work within a formal system, but simply in “naïve” arithmetic.
Nor did Cantor work in an axiomatic framework.
Drake (Set Theory (1974)), dismisses the set-theoretic systems (NF, ff.) of the towering Quine, on the grounds that they are not based on an intuition of what sets really are, being more in the nature of formal exercises, “and thus, not a set theory in our sense at all”. -- Take that, Quine. You write like an angel, but you reason like an atheist. (For which “nominalist” is the polite name…)
Thus: Set-theoreticians are attempting to explore a perceived reality. It turns out to be too full of counter-intuitive features to allow us to proceed forever informally, so we axiomatize. It is too vast to fit into any single axiom system; we use different systems, depending on what kind of big game we’re going after. (Much as, in problems of physics, we may be content with a classical approach, or, depending on the problem, may need to bring in the quantum or the relativistic.) If it’s very big, we add Choice – a sort of elephant gun that threatens to explode in our faces. If we wish to do more than to interview the first set that we meet on the street, and do a census of the whole ontology, we add axioms of infinity, of various strengths. It’s a lot like people pottering around in a lab.
Once added, the axioms serve, not as a guide, but more like a guard-rail. To make new discoveries, we still rely upon intuition (whatever that may be – like free will, it is blazing, surprising). Eventually, these explorations may be solidified in the acceptance of a new axiom into the canon.
Bertrand Russell famously quipped:
The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil.
This witticism has a nonzero, yet limited, domain of relevance. (Lord Russell’s shafts are barbed, but brittle, and do not sink deep. Moreover, they may boomerang. Geach: “Russell’s Axiom of Infinity lies open to his own taunt about the advantages of theft over honest toil.”)
We would no more lightly add an axiom to our system, than a tenet to the Credo. Any new axiom must (on the positive side) “play well with others”, yielding in concert with its mates, new results otherwise underivable, but which, on other (intuitive – one might say, mystical) grounds, we believe to be true; and (on the negative side), not give rise to paradox or contradiction. Such helpmates do not lie ready to hand. (Wang? Levine?) has stated:
The apparent hopelessness of finding new axioms has become a source of scepticism about the theory of the infinite.
Once again one is led to reflect, how far in advance of his time was Euclid – by millennia, maybe. Not only the axiomatic method überhaupt, but the nice intellectual scruples that led the Euclideans not simply to accept the Parallel Postulate (though it had proved its fruitfulness in countless theorems), but to attempt to eliminate it as an axiom -- to demote it from axiomatic status -- by derivation from the other axioms. Only when (much) later thinkers had developed concrete models showing consistent geometries in which the Parallel Postulate, so far from being logically superfluous because derivable as a theorem, was actually false, was the attempt abandoned; and only then did men venture to canonize axiomatic alternatives to that Postulate (resulting in Riemannian and Lobachevskian geometries respectively).
It’s not that you simply posit the complex plane, like some sort of board game, and see what happens. The complex plane fits the facts. It has jobs to do, antecedent to its formal positing.
So: Despite the logical and rhetorical set-up, of starting from the axioms, and reasoning one’s way to lemmata and theorems, the historical record is rather the reverse: First the observations, then the axioms.
*
This practice, of judging the roots by the fruits, is called by Michael Potter the “regressive strategy” (Set Theory and its Philosophy, p. 34). He quotes Weyl to the effect that “this attitude is frankly pragmatic”, and adds a caveat (p. 220):
Regressive arguments for any set-theoretic axiom depend on a prior belief in the mathematical truth [[emphasis in original; he means, as opposed to some technical, theory-internal matter restricted to set theory]] of some consequences of the axiom, but the fact that they are consequences of it depends in turn on an embedding of parts of mathematics in set theory: a different embedding may not require the same axiom, and so the regressive justification is relative to the embedding.
Compare Kolmogorof (Mathematics III.142):
The concept of axiom is relative: One and the same statement can emerge as a theorem in one buildup of a theory, and as an axiom in another.
And Boolos (1971, quoted in Potter 2004, p. 297):
The reason for adopting the axioms of replacement is quite simple: they have many desirable consequences, and (apparently) no undesirable ones.
(The Boolos of (2000) is apparently sadder but wiser, expressing “at some length his discomfort with the ontological commitments” of Replacement.)
At its worst, such a strategy could devolve into “Whatever works”. At its most sophisticated, it represents a special case of the Duhem-Quine thesis, central to modern thought, whereby the propositions of any theory face the tests of reality, not individually, but as a corporate body. The theories in question are normally conceived of as those of a science, such as physics; but the insight applies as well to our workaday “theory of everyday life”.
For an example: Potter writes (p. 300)
What we have shown is that the conjunction of the following three assumptions is contradictory:
(1) second-order logic
(2) Basic Law V;
(3) the assumption that there is a single domain of objects over which all quantifiers range.
But we should not leap too hastily to judgement as to which of the three is guilty.
In the Duhem-Quine perspective, even that is a rush to judgement. For this is not a game-show, on which we must open precisely one of three doors. Quite possibly our eventual solution will be one which involves ideas from each of the three, but in which none of the three survives intact, as such.
*
Potter adds (p. 251):
Many authors have taken the fruitfulness of an axiom as an argument for its truth. Curiously, though, one occasionally finds the opposite view expressed: “The more problems a new axiom settles, the less reason we have for believing the axiom is true.” (Shoenfield 1977).
[Note: The term “regressive” seems to be nonstandard. Cf. Wikipedia on the same or similar idea, with different terminology:
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. The method can briefly be described as "going backwards from the theorems to the axioms."]
Though admittedly pragmatic, the strategy aims at helping to tidy up the abstract axioms. I would further add that, owing to the peculiarly practical constitution of the human mind, we need frequently to refresh ourselves with such regressive moves, if the axioms and other basic principles are to make any sense to us at all. To understand what such a statement really says, we need to understand how it is motivated.
This fact is particularly important in mathematical and scientific exposition. The crystalline publications of Gauss are notoriously hard to penetrate, since he was “the fox who erases his tracks with his tail”. To his frustrated critics, he replied with scorn: When once you have constructed the building, you remove the scaffolding. But: If, as is frequent in theoretical physics and in higher mathematics, there are no doors or windows on the ground floor, we need the homely practical scaffolding to clamber up and get inside.
To take a concrete and more recent example:
My bookshelf has long groaned beneath the weight of a standard work by Eilenberg and Steenrod, Foundations of Algebraic Topology. Periodically I return to the assault of its north face, and am always hurled back defeated. And this, apart from the inherent difficulties of the subject, because Foundations is here the operative word. As the authors state their goal (Preface, first paragraph):
The principal contribution of this book is an axiomatic approach to the part of algebraic topology called homology theory. … The present axiomatization is the first which has been given. The dual theory of cohomology is likewise axiomatized.
The reader is forewarned (p. x):
No motivation is offered for the axioms themselves. The beginning student is asked to take these on faith [emphasis added] until the completion of the first three chapters. This should not be difficult, for most of the axioms are quite natural, and their totality possesses sufficient internal beauty to inspire trust in the least credulous.
For example, among the Axioms for Homology, we find the Exactness Axiom (p.11):
If (X,A) is admissible an i:A => X, j: X => (X,A) are inclusion maps, then the lower sequence of groups and homomorphisms
i* d j* i* d
..<= Hq-1(A) <= Hq(X,A) <= Hq(X) <= Hq(A) <= …
is exact.
Now, who could quarrel with that?
The book is a valid piece of work, an acknowledged classic; but, despite their casual reference to “the beginning student”, it is not a pedagogical work; it must be understood regressively, if at all. And indeed the authors are well aware, that in any splendid mathematical edifice, the foundations are the last to be built. For, homology is “the oldest and most extensively developed portion of algebraic topology”; they have been involved with it the whole of their professional lives. They have long used it intuitively, and now, retroactively, they intend to formalize. It is in no way a subject that is given a priori – its truths may abstractly enjoy that status, along with other statements of mathematics, but as the authors put it (p. viii):
A picture has gradually evolved [emphasis added] of what is and should be a homology theory. Heretofore this has been an imprecise picture, which the expert could use in his thinking, but not in his exposition. A precise picture is needed. It is at just this stage in the development of other fields of mathematics that an axiomatic treatment appeared and cleared the air.
Compare:
The axioms codify ways we regard mathematical objects as actually behaving. … The role of axiomatics is largely descriptive. A Foundational system serves not so much to prop up the house of mathematics as to clarify the principles and methods by which the house was built in the first place.
(R. Goldblatt, Topoi , 2nd edn. 1984, p. 14)
This service of codification, however, is accessible only to those who have paid their dues.
Note: Eilenberg himself acknowledges the cognitive difficulties of their approach:
Algebraic topology is … at a first approach, a bewildering field. First, the tools used sometimes look weird … A further source of obscurity is that these tools are usually studied before the problems to which they are to be applied are even mentioned.
-- Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 98
Despite myself, I am reminded of the plight of the Koran-school pupil, memorizing before the language of classical Arabic has been mastered (if indeed it ever is).
*
A similar situation obtains in theoretical physics.
Ph. M. Morse & Herman Feshbach, Methods of Theoretical Physics (1953), p. 266:
A new equation for the description of new phenomena is seldom first obtained by strictly logical reasoning from well-known physical facts; a pleasingly rigorous description of the equation usually is evolved only about the time the theory becomes ‘obvious’.
(Note: ‘evolved’ rather than ‘deduced’.)
And R. Adler, M. Bazin & M. Schiffer, Introduction to General Relativity (1965), p. 32:
Historically, the notion of contravariant and covariant representations was naturally introduced by generalizing this particular case of vectors in a Euclidean space, and not in the axiomatic way which we choose to follow in this chapter.
*
André Weil was one of the pioneers of algebraic geometry -- I almost wrote “founders” but pioneer gives a better sense of the exploratory forays that his early activity involved. Before long, though, he too wrote a Foundations books -- the Foundations of algebraic geometry (1946). In the Introduction, he gives a glimpse of the exploration/formalization/re-launching dialectic :
The so-called “intuition” of earlier mathematicians, reckless as their use of it may sometimes appear to us, often rested on a most painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed …
(That is the phase recounted in our
parable of Wisedome Woodchuck, Apostle to the Groundhogs, who dimly intuited Commutativity, before the idea had been formalized.)
But come formalization time, fun’s over:
Our method of exposition will be dogmatic and unhistorical throughout …
Yet, this necessary and aseptic task accomplished, the author reverts to his free-wheeling persona, in an almost lyrical outburst:
… to the reader, to whom the author, having acted as his pilot until this point, heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea.
*
Consider, indeed, the parallel with the role of a Constitution in the affairs of state.
Our own was drawn up, not by political scientists, but by the veterans of the anti-colonial struggle and eventual Revolution. The framers had paid dues. The success of the document – which has been substantial – is due partly to this, and partly to the dynamic balance between the inherent conservatism of a Constitution (much like an axiom system), and its ability to evolve in response to events. The original document was a bit like ZF, so to speak – solid but a little dry; then the pithy and powerful Bill of Rights was added (rather like the “C” in ZFC).
Transplanted to foreign soils, however, where it has not grown up, but only been thrust into the ground, the organism wilts and dies. Many a tyranny has drawn up a Constitution that looks perfectly splendid under glass.
(Cf. Tocqueville I.viii:
La constitution des Etats-Unis ressemble à ces belles créations de l’industrie humaine qui comblent de gloire … ceux qui les inventent, mais qui restent stériles en d’autres mains.
C’est ce que le Mexique a fait voir de nos jours.
Mexico copied the U.S. constitution litteratim, yet:
Le Mexique est sans cesse entraîné de l’anarchie au despotisme militaire.
This was published in 1835, and it still applies.)
*
One learns to beware of books with Foundations… or Grundlagen… in the title. To the unwary, these might sound like “Introduction to…” or even “… for Dummies” (“Foundations for a Healthier You”). Not a bit of it.
It is not readily apparent, in a mathematical context, just how abstract and removed a foundational treatment is. For: pre-axiomatized algebraic topology was already complex enough; and the proposed set-theoretic foundations for mathematics look pretty mathy themselves; so you figure they are in the same line of work.
The difference in kind between the two bodies of doctrine – the founding and the founded -- becomes more apparent if we look to a field with more intuitive content: mechanics. And not quantum mechanics either – just good old orbiting planets and billiard-balls.
There is a celebrated volume by Ralph Abraham (1967; 2nd edn. 1978) called Foundations of Mechanics. It is beautifully bound, printed on thick creamy paper, and sports a gallery of full-page photographs of the tutelary deities of the subject, from Galileo and Kepler to Al Kelly and Steven Smale. The message: A jubilee monument, to a field that has truly come of age.
Topics include: Banach spaces, vector bundles, Cartan’s calculus of differential forms, symplectic geometry, etc. Topics do not include: Anything you know about. It is not until the very end of the book that we get a hint of the possible existence of an actual physical world for all this to apply to. Indeed, the sense seems to be that, since we have derived it all so beautifully, the universe itself is more or less de trop.
The comparable case in religion is: Theology and the credo as – latter-day, retrodicted – underbuttressings for religious experience (which is spontaneous, and prior).
*
Interlude on the logical status of free will.
(1) By freedom of will, I mean only and exactly that: the freedom to will something. You might be unable to do that something – you might, in fact, be completely paralyzed, deaf dumb and blind, or a brain in a vat. The will remains, until you are yourself extinguished.
Compare this coffee-cup: Though it yearns with every atom of its being to unite itself with the center of the earth, it is prevented from so doing, by the counterforce of this desk. The gravitational field nonetheless exists, and is effective.
(2) Since I can see it with my eyes closed, and touch it and taste it without moving, I am more certain of my free will than of anything; in particularly, moreso than of yours. Psychologically, I am about equally certain; but logically, to conclude to the existence of someone else’s free-will – to the existence, indeed, of Other Minds -- obliges one to an additional metaphysical step, perhaps to be bridged by the addition of another axiom. It is a step which I am happy to take, since it is basically the same pons asinorum over which we must pass, to conclude – for example -- to the actual solid existence of this coffee cup which I am, to all (potentially deceptive) appearances, right now holding in my (not to beg the question of its own vexed hypothetical existence, but for the sake of exposition, let us so denominate it: ) hand. Namely, that of: Der Herrgot is rafiniert, aber boshaft ist er nicht. So much follows from that.
The quotation is from Einstein, and it guided – not led, but guided, again like a guard-rail – his work in physics. Its English translation supplied the title for the classic Einstein portrait by Abraham Pais, Subtle is the Lord (…”but He is not plain mean” – that is, he wouldn’t create a world that is deliberately, perversely misleading).
*
Now: Once the axioms have been (not posited ex nihilo, but) extracted from practice, they might manage to be profitably re-applied in some other direction. This smacks of the dialectic; thus it is appropriate that an example be taken from the 1956 Soviet anthology (US title: Mathematics: Its Content, Methods, and Meaning), in which Kolmogorov writes (II.253):
Probabilistic methods have proved to be applicable to questions in neighboring domains of mathematics, not “by analogy”, but by a formal and strict transfer of them to the new domain. Wherever we can show that the axioms of the theory of probability are satisfied, the results of these axioms are applicable, even though the given domain has nothing to do with randomness in the actual world.
This is an instance of the “curious portability” of mathematical principles.
*
To return to our point: The Axiom of Choice is not derivable from ZF, but has an independent sense, and, in tandem with the more basic axioms, has all sorts of mathematical consequences. Free will is (so far at least) not derivable from physical science; and from this fact, certain materialists have loudly concluded to its non-existence – a fallacy, as the set-theoretical case by analogy suggests.
Similarly: The Riemann Hypothesis may be true, in which case it is a necessary truth; or it may be false. And the question may never be settled, this side the grave. We do not therefore reject its investigation, the way some dismiss out of hand the mere consideration of questions of theology.
A note: Whatever we might (doubtfully, and fallibly) derive further, the axiom system is not intended as a mere formal exercise, in the sense that it hardly matters what we toss in or toss out; nor should it be controversial. Before we go further: It is necessary to check for compatibility of axioms, yours and mine, if any fruitful conversation is to take place. So, nota bene: If you deny Choice (that is, free will), then here we really must part company. For you cannot honestly disbelieve it. If you say that you do, or think that you do, then it is either because you once took some stupid Intro-to-Phil course for a distribution requirement, and swallowed whole what the professor peddled (and which he himself did not believe, though he may drink quite a bit after class to dull his conscience, and to wash out the foul taste of the tripe he’d been spouting), or else because you yourself are a professor of philosophy, probably a bit on the odd side sexually, and disinclined to return items you have borrowed, holding court at some atheist joint, and paid by your paymasters to contrive ingenious paradox, to baffle and belittle the common sense of the janitor, who is paid less.
Am I becoming abusive? Yes.
*
Though the above is a satirical riff, I do rather mean it. For, whoso should seriously deny, nay indeed literally disbelieve, the existence of free will, lives by a doctrine that puts him at variance with all normal human relations. Thus, should his humors and hormones come to slosh into a configuration whereby (spotting that tasty specimen over there) they are fain to rape it and then kill it, or to kill it and then rape it, who is there to say nay? Nobody.
The materialist replies (with a hint of a blush): Just because I think I’m an automaton, doesn’t mean I necessarily have to do bad stuff…
Oh, come on. If you’re a robot, you rampage. That’s what robots do. Haven’t you read Dilbert?
*
So, we take free-will as axiomatic, and feel quite justified in so doing.
Quite other is the case of belief in God. It has been an axiom for many thinkers throughout history, but its logical/experiential status differs. Here is a thesis on which we may really and sincerely disagree -- disagree even with our own selves, on different days -- yet the conversation can continue. For, His existence may (or may not!) turn out to be a “necessary truth”, but that by no means renders its truth or falsity, or even its meaning beyond vague intimations, self-evident to the senses, any more than the necessary truths of Algebraic K-Theory are. We discover Him (if we do at all), bit by bit, in curious encounters such as certainly admit of alternative interpretations, and suggestive of contradictory results. On occasion, as in Set Theory, we keep barking our shins upon paradox, which may give rise to some serious theological contortions (compare, in Set Theory, the Theory of Types) to explain away. You are no more guaranteed to get a handle on Him (and His choir of angels), even after a lifetime of quest and study, than you are (born in a hovel in Africa or Cleveland) to claw your way out of the surrounding intellectual dreck and get clear on E8 (with its cohort of angles), or to re-discover the Riemann Hypothesis, let alone settle it.
Some of my favorite acquaintances have been observant Jews of extensive scientific training, who confessed themselves intellectually agnostic. Quite an honorable stance – Einstein felt the same way about Quantum Theory. Bishop Berkeley, for his part, acknowledged the Godhead, but he wasn’t going to swallow those newfangled Newtonian fluxions without some thoughtful chewing.
For this reason we do not consider an axiom ‘G’ (existence of God), instead of C, to supplement our materialist/formalist ZF. G would have been less surprising, given Western intellectual history, but C is empirically more parsimonious.
*
So, a minimal set of axioms. Our proposals are in the spirit of Chesterton’s essay “The Diabolist” (in Tremendous Trifles), p. 101:
“Aren’t those sparks splendid?” I said.
“Yes,” he replied.
“That is all that I ask you to admit,” said I. “Give me those few red specks, and I will deduce Christian morality."
*
Quine, re “the alien terms of the annexed lobe” (don’t ask), remarks:
It is as if some scientifically undigested terms of metaphysics or religion, say ‘essence’ or ‘grace’ or ‘Nirvana’, were admitted into science along with all their pertinent doctrine, and tolerated on the ground merely that they contravened no observations.
That is, in our terms, that neither their assertion nor their denial were derivable from the core axioms (our “ZF”).
I quite agree with that veteran nominalist, that those terms are ill-suited to adjunction as axioms: unlike free-will, which is an empirical given if anything is (“Cogito, ergo sum”). To seek again a set-theoretic analogy: One would not wish to adjoin the continuum hypothesis as an axiom to set theory, even though (as Kurt Gödel and Paul Cohen jointly proved) you would get into no ‘observational’ difficulties by so doing. (Specifically, its truth or falsity is independent of ZFC.) For, the continuum hypothesis was in its origin taken to be a matter of fact – or of falsity -- rather than a parameter of theory. That is: There either is or there is not, one would think, a subset of the real numbers, not equinumerous with either the integers or the reals. If it exists, we wish to examine it, turn it over in our hands, learn more about it. It is not a thing to be simply posited; and we should be very disappointed were its existence to be established merely by some sort of wretched diagonalization argument, that gave no hint or glimpse of its anatomy. That the hypothesis proves independent of ZFC is startling. We still feel there should be some “fact of the matter”, but now realize (sadder and wiser), that it will be a relative thing – true in this extended system, false in that. (The more nominalistically minded would say, given the independence result: Give Up, there *is* no fact of the matter; but Gödel, a Realist, was not satisfied with that. See his argument Contra Errera in “What is Cantor’s Continuum Hypothesis”.) So likewise: God, and grace, and the afterlife, are things to be somehow learned about (positively or negatively), not posited for free.
To be distinguished from axioms, are working assumptions. These may be quite as essential to our everyday thought and action, only their logical status differs. I mean such metaphysical principles as causality, induction, and the goodness of God. We need these notions in a practical sense, if we are to go about our affairs. It may be that some levels of the physical world are in fact acausal; that the rough-and-ready, Hume-maimed principle of induction, is indeed but ready and rough; or that the Creator is so remote from us, as to be morally and humanly inscrutable. The sane man expends little energy worrying about such possibilities during normal working hours.
*
This whole post is, to be sure, something in the nature of a Gedankenexperiment, or finger-exercise, or even a sotie. I do not seriously mean to approach human reality axiomatically. But this is not owing to any doubtfulness about free-will, or about rationality (including theology) as a legitimate field of inquiry. Rather, the axiomatic method has been pretty sterile even in physics; and in math, of little import outside of strictly foundational subjects like Set Theory, and these subjects have proved less fruitful in mathematics at large, than had once been hoped. The point is simply, that we do as a background fact tacitly assume the existence of some sort of framework of basic principles, of scientific hue; and that we could legitimately strengthen the system by the adjunction of a judiciously chosen axiom lying outside their range. Indeed, it’s difficult to think what scientific principles have anything like the stability or certainty of the existence of our free will: the only physical concepts to have survived intact into quantum theory, one reads, are Entropy and Action, both quite abstract. And as for Relativity: Einstein out-relativized Galileo, and was universally embraced; yet now one reads of motion relative to “the fabric of Space”. What’s a fellow to believe?
Believe the Creed, and leave the rest to fate.
