The Vacuous Identity Principle
Constructivism without Constructivism
While I was recently in Uruguay I wrote a post on constructivism – Constructivism without Constructivism – in which I proposed a way of thinking about constructivism that I have since realized can be generalized as a distinct principle of reasoning.
We all know what it is like to “agree in principle” even while continuing to disagree over facts and details, and indeed disagreeing about everything other than the principle under consideration. This was the approach that I suggested for thinking about constructivism: that we might agree in principle that imposing limits on formal thought in order to ensure the consistency and coherency of such thought, even while not agreeing to the particular limitations proposed by contemporary constructivism.
Constructivism and Kantian Discipline
Constructivism seeks a more disciplined approach to formal reason than that of classical eclecticism (the latter term I have taken from Torkel Franzén). To speak of discipline in connection with reason immediately suggests Kant’s characterization of discipline near the end of the Critique of Pure Reason:
“The restraint which is employed to repress, and finally to extirpate the constant inclination to depart from certain rules, is termed discipline.”
But to understand the formal discipline advocated by constructivism through the lens of Kant’s conception of discipline is problematic, since constructivistic thought itself can be understood as a deviation from classical rules of logic (intuitionistic logic is sometimes identified as a deviant logic), so that discipline might be required to restrain the impulse to depart from classical rules of reasoning (such as tertium non datur, or P or not-P).
Yet the spirit of Kant’s conception of discipline, and its essentially constructivist character (in harmony with Kant’s oft-noted proto-constructivism), is revealed in the immediately preceding sentence to that quoted above:
“…where the limits of our possible cognition are very much contracted, the attraction to new fields of knowledge great, the illusions to which the mind is subject of the most deceptive character, and the evil consequences of error of no inconsiderable magnitude—the negative element in knowledge, which is useful only to guard us against error, is of far more importance than much of that positive instruction which makes additions to the sum of our knowledge.”
The constructivist’s focus on limiting classical eclecticism to a more disciplined subset of formal reasoning techniques is clearly an instance of, “the negative element in knowledge,” and, in so far as discipline must be employed in order to extirpate the tendency to depart from the constructivistically acceptable rules of formal reasoning, just so far is constructivism disciplined.
Constructivism and Cantorianism
Given that limitation is the essence of constructivism, it is no wonder that Cantorianism is the bête noir of constructivists, with Cantor’s transfinite numbers as symbolic of the epistemic hubris of non-constructive thought that recognizes no limits, no boundaries, and no intrinsic finitude to human cognition.
Set theory and transfinite numbers have long been singled out for particular execration by constructivists. Cantor himself was personally singled out. Kronecker called him a “corrupter of youth,” which puts Cantor in the excellent company of Socrates. Cantor felt the opprobrium leveled against him no less than did Darwin, not withstanding Cantor’s sincere piety and his sincere efforts to make his thought theologically palatable. Both Darwin and Cantor knew that they were offending to the intellectual pieties of their time, even as both transcended the limitations of their time.
Despite the great many differences among constructivists themselves, and the many paths proposed to constructivism, all can agree that Cantorianism is beyond the pale. Intuitionists, predicativists, finitists, all hold in common their rejection of infinitistic reasoning. (Just as many of Darwin’s contemporaries agreed in principle about evolution, but rejected the mechanism of natural selection.)
Do constructivists really completely reject Cantorism? The story of constructivism since its inception has been a story both of the growing sophistication of constructivist methods, and the growing number of non-constructively discovered results that can be reproduced by constructive methods. But I will leave this aside for the time being, perhaps to recur to this interesting observation at some future time (a time not specifically identified and therefore a non-constructively identified time).
The Constructivist Consensus
Given that the many different schools of constructivism, which are legion, and which are, many of them mutually incompatible, agree in principle that non-constructive and infinitistic reasoning is fatally flawed, there is something more to constructivism that mere limitation -- or perhaps I should say that constructivism is essentially concerned with the kind of limitations to formal thought that yield finitistic reasoning.
Well, I can even go this far in agreeing in principle with constructivism. Logic and mathematics must be thought by finite human minds, and yet I still reject that particular constructivist constraints placed upon formal thought.
Introducing the Vacuous Identity Principle
This possibility of agreeing in principle without agreeing in fact I will call the Vacuous Identity Principle. When two or more individuals agree upon a principle, but agree about nothing other than the bare principle itself, they accept the identical principle, but without instances in common the principle is vacuous.
From the standpoint of pragmatism, the vacuous identity principle is no principle at all. If in logic and mathematics, as elsewhere, we are to observe the principle by their fruits ye shall know them, and the formal fruits of two logicians or mathematicians who assent to the vacuous identity principle but agree on nothing else, remain distinct, then we cannot “cash out” the identity of principle.
To turn this around, we can see how much of our thought, even in purely formal matters, is pragmatically driven, given that a purely formal assertion of agreement in principle may have no practical consequences.
Vacuous Identity and Vacuous Distinction
In formulating the vacuous identity principle I realize that I have previously come across the contrary of this principle, and this was an assertion by J. L. Austin that, “A distinction which we are not in fact able to draw is — to put it politely — not worth making.” (a line I also quoted in my post Of Distinctions, Principled and Otherwise)
These principles -- my Vacuous Identity Principle and Austin’s rejection of vacuous distinctions -- are two sides of the same coin, and if we recognize the validity of the vacuous identity principle, we also ought to recognize the validity of distinctions that make no difference, i.e., that have no practical consequences -- in other words, the Vacuous Distinction Principle.
The vacuous identity principle is a principle of generalization, while the vacuous distinction principle is a principle of formalization.
An Indispensability Argument for Vacuous Concepts
An identification without identity is an identification not worth making, one might say, as one might also say that the distinction without a difference is not worth making -- except that I think it is worth making. Thus in asserting the Vacuous Identity Principle I am asserting the legitimacy of vacuous concepts, and I think this is important to recognize, since vacuous concepts often play a significant role in formal reasoning.
The distinction made between a unit set and its only member is dismissed by some as a vacuous distinction, and an overly-subtle distraction in set theory, but I don’t see it like this at all. I think it was Frege who pointed out that the concept “natural satellite of the Earth” is a set with one member, while the concept “natural satellite of Venus” is a set with no members, and therefore, in a sense, a vacuous concept -- but a perfectly legitimate concept.
Any concept of which we can predicate the number zero is a vacuous concept, but contemporary mathematics could not get by without the concept of zero. Therefore there is an indispensability argument for vacuous concepts, and therefore also, presumably, for both vacuous identities and vacuous distinctions.
