Definitions are critical in this discussion, specifically definitions for possibility and conceivability. In the paper, possibility is plain-old metaphysical possibility. The definition of conceivability is the main focus of Bryce's work. The definition he gives is Wittgensteinian:
The act of conceiving must entail at least the ability to list out the conditions of the state of affairsʼ fulfillment. If these conditions cannot be expressed, then we cannot be sure that what we take to be conceived of is actually conceived of, for understanding what would be the case just goes along with knowing what it is that is the state of affairs in this case.For a more concrete example, he considers the proposition that a specific polar bear is blue rather than white. The way we would know whether this is the case is by looking at the polar bear and seeing what color it is. This leaves open the obvious objection that a blind person cannot see the polar bear and thus cannot conceive of the proposition in question. To this Bryce replies that he does not mean conceivable by anyone, but only by those with sufficient knowledge to conceive of it. While the blind person cannot conceive of color, someone who can see can so conceive, and that is enough.
It is here at the beginning that we run into the first major problems with Bryce's thesis. Suppose we are going to conceive of proposition P. In order to do this, we must know the conditions of P, Q and R, both propositions. And how do we conceive of Q and R? Obviously, by knowing the conditions for them, Q1, Q2, R1, and R2. But the problem remains and has even multiplied. Thus, in order to conceive of a single proposition, we must be able to conceive of many more with no obvious stopping point. Where does conception end? If all propositions are self-conceived (whose condition is the proposition itself) or analytic from self-conceived propositions, then we have no problem. (We need analyticity because we can't have any notions or relationships that we can't conceive of already, so synthetic propositions are out.) This gives us what we want, since it means that all true propositions are necessary, and necessity implies possibility, but this isn't what Bryce is arguing.
The notion of self-conceived propositions raises an issue all by itself. To use Bryce's example, what are the conditions for the proposition that a polar bear is blue? The answer he gives is our seeing the polar bear and its color, but that's not actually correct. Our vision tells us what color the polar bear is, but it does not determine the polar bear's color. Rather, the condition for the polar bear being blue is the fact itself, not our knowledge of this fact. Thus, to conceive of a proposition is to understand its meaning, and Bryce's claim transforms into a proposition is possible if and only if it is meaningful, which isn't quite the same thing.
Alternatively, we might suppose that the conditions for empirical propositions, like the polar bear's color, are all the prior events which caused the polar bear to be blue. But this means we have to be able to list every event and causal connection in a given chain going back to the first cause, if there even was such a thing. This leads to the absurdity that in order to conceive of my writing this essay now, I have to know everything in the causal chain that led up to it, which neither I nor anyone else does or can. Thus, it is impossible that I am writing anything.
Bryce makes brief digression into the difference between conception and imagination. This is probably the weakest part of the paper, since at the end we still don't have a very good notion of imagination. Near as I can discern, the main difference between imagination and conception is that imagination doesn't require grounding in logic. In a sense, imagination is precisely what it says: forming an image of something. The problem is that all of Bryce's criticisms of conceiving something impossible apply also to imagination. This seems to be a relatively un-worked-out part of Bryce's system, so I won't dwell on it, but imagination is definitely something to elaborate further.
The long section entitled "Conceivability Implies Possibility" is quite fascinating. I'm quite impressed with Bryce's work here, and my only difference would be to question the need establish the identity of different "nothings". I am perfectly happy saying that the set of square circles is the same as the set of triangular circles, both empty, and don't see the need to descend to first-order logic. I suppose a metaphysician like Bryce might have some reason to do so, but logic and mathematics are my bailiwick, so I will leave this matter of esoterica to the specialists.
My focus on mathematics does allow me to spot several serious problems that emerge when we try to apply Bryce's system to mathematical propositions. Up to this point, we've only dwelt on problems that require significant revisions to the overall thesis. Now, however, we encounter problems that actually prove that Bryce's system simply does not work.
The standard for conception of mathematical propositions that Bryce provides is rather high:
To conceive of a logico-mathematical proposition is to know that it is true; the demonstration of its conceivability is the same as the demonstration of its actuality.That is, we can only conceive of a theorem if we can prove it. This is obviously problematic in that it means we can't conceive of a theorem that we haven't simply proved yet, making it impossible. In other words, Fermat's Last Theorem was impossible twenty years ago, but Andrew Wiles somehow made it true in the intervening years. The way Bryce tries to escape this problem is by proposing agnosticism on the conceivability of unproven theorems--we simply don't know everything that we can conceive. However, this doesn't really solve the problem since in actual fact we either can or cannot conceive of a theorem. If it is the case that we can conceive of a theorem, then we can prove it. If we are charitable, then we might say that we can prove theorems that we merely haven't proved yet. The task of mathematicians is to work out the true extent of our powers of conception.
This "solution" rests on a rather naive idea of mathematics as an essentially unchanging body of knowledge--all mathematicians ever do is work out further implications of various axioms. We can forgive Bryce for this mistake since mathematics isn't his specialty, but the implications are still quite serious for his theory. There three different ways this proves to be a difficulty.
For one, mathematics today is very different from mathematics even two hundred years ago, let alone two thousand. In Socrates's day, no one could prove Fubini's theorem or derive the Cauchy-Riemann equations, however hard they thought--they simply didn't have the necessary precursor notions. Completely axiomatized mathematics is actually a very modern invention--we've only had axioms for arithmetic for a little over a hundred years now. This is a problem for Bryce's system in that we today can prove theorems which were unprovable two thousand years ago and not because we've worked out the consequences of our axioms better--it's instead because we have axioms where people long ago did not. We can avoid this difficulty if we posit strong formalism; that is, Theaetetus could have proved all the theorems of group theory had he our axioms, but since those axioms don't actually describe anything, he wouldn't have any better understanding of anything. However, strong formalism is absurd on its own terms, and Bryce's, as we'll see in a moment.
Even if we restrict our concept of mathematics to axiomatic systems, we still have to deal with the issue of picking axioms, even to describe the same subject. If we are going to do set theory, we have a veritable smorgasbord of options. We can have sets alone, proper classes if we'd like, and this is leaving aside constructive set theory (which isn't really axiomatic) and category theory, among others. Which do we pick? Once more formalism saves us, but it really is no help. We saw earlier how we could derive from Bryce's general thesis that a proposition is possible if and only if it is meaningful, and even if we don't take this tack, I doubt Bryce would contend that we can conceive of meaningless propositions, even if they are not logical impossibilities per se. But strong formalism contends that all of mathematics is meaningless, and thus inconceivable and impossible. I'm afraid I can't see any interpretation of this other than that mathematics is utter nonsense, and I invite anyone who thinks this to start counting the words on this page and then explain what it is they are doing.
There is a third problem that formalism cannot allow us to escape: unprovable theorems. One example of an unprovable theorem is the Continuum Hypothesis--that the cardinality of the continuum is the smallest uncountable infinity. We know that this cannot be proved or disproved using the present axioms of set theory, so does that mean we cannot conceive of it as true or false, making both impossible? That is the answer Bryce's system seems to require. Another example would be in arithmetic. We know that for any countable set of axioms and proof methods, there will be theorems which are true under those axioms but that cannot be proved from those axioms. Does this mean that these theorems are inconceivable and impossible? That's what the theory says, but Goedel proved that they do exist. Thus, Bryce's theory is wrong.
As I had hoped, I've come up with a few ways that Bryce might take to save his theory: (1) replace conceivable with meaningful; (2) replace conceivable [by humans] with conceivable by an infinite mind; and (3) devise a new, non-Wittgensteinian notion of conceivability. Any one of these changes would sufficient to deal with the flaws I've pointed out, though they would still need elaboration. However Bryce continues from this point, whether in a direction I've suggested or not, I wish him luck.
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