"infinity is a concept" -> practically useless, requires clarification and usually used as placeholder for "infinity is not a number"

"infinity is an equivalence class" -> mumbo jumbo, but mumbo jumbo we can explain.

the reason i like thinking about infinity as an equivalence class is that it allows reasoning about it easier. The equivalence relation is context dependent but infinity is also context dependent so that doesn't change much.

For example:

1. some infinities are larger than other infinities -> some members of (things we call) infinity have higher cardinality than other members

2. infinity+1=infinity -> infinity is closed under addition by finite numbers.

In my recent post Precise Generality I promised that I would follow up with further exposition. This post is not that promised follow-up providing further exposition and investigation of the idea of precise generality, but truly an “addendum” simply to add some further detail to the ideas expressed in Precise Generality. In that post I wrote:

“In so far as a formal system is the only known embodiment of a precise idea of precision (according to Heyting), and in so far as a formal system is intrinsically finitistic (in both conception and execution), the only precise idea of precision we possess is intrinsically finitistic.”

It is interesting to explore the ideas that Gödel had regarding formal systems in relation to their finitary nature. On the one hand, Gödel seems to unambiguously embody the above idea of formal systems being intrinsically finitistic. In particular, Gödel cited the work of Turing with approval:

“In consequence of later advances, in particular of the fact that due to A. M. Turing’s work a precise and unquestionably adequate definition of the general notion of formal system can now be given . . .”

Gödel added a footnote to this: 

“In my opinion the term ‘formal system’ or ‘formalism’ should never be used for anything but this notion.” (Collected Works, Vol. I, p. 195) 

More generally regarding formal systems Gödel wrote:

“A formal system can simply be defined to be any mechanical procedure for producing formulas, called provable formulas.”

Gödel goes on to say that the syntactical characterization of formal systems which he gives at the beginning of his 1934 paper (“On undecidable propositions”) is equivalent to the formulation in terms of mechanical procedures if his use of ‘finite procedure’ is understood to mean ‘mechanical procedure.’ Gödel then reiterates his point by mentioning, “...the concept of formal system, whose essence it is that reasoning is completely replaced by mechanical operations on formulas.” He also notes that the possibility of finite non-mechanical procedures such as “the use of abstract terms on the basis of their meaning” does not detract from the adequacy of the suggested definition of “formal system.”

But is not the adequacy of mechanical procedures for finitary reasoning relative to a conception of proof, so that if a broader conception of proof which takes account of human motivations (such as, e.g., the desire to rigorously formalize as much of classical mathematics as possible) is at issue mechanical methods are no longer adequate? If one assumes a notion of proof reducible to mechanical procedures and goes on to prove it so, this is nothing but a petitio principii. Gödel cannot be accused on this account, as he did not anticipate Turing’s work, though he may have reinterpreted his earlier work in the light of Turing’s results, and so engaged in a kind of personal anachronism.

Despite Gödel’s apparent willingness to revise his own thought in the light of Turing’s work, Gödel was in no sense limited by the influence of Turing, and in fact in one of my favorite quotes from Gödel (which I have quoted in many posts), he explicitly takes on Turing’s finitism:

“Turing... gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclusive. What Turing disregards completely is the fact that mind, in its use, is not static, but is constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure.” (“Some remarks on the undecidability results” in Gödel, Kurt, Collected Works, Volume II, Publications 1938-1974, New York and Oxford: Oxford University Press, 1990, p. 306)

There is, of course, no contradiction, not even any paradox, in observing that formal systems are finite but that the mind (which thinks formal systems) converges upon infinity. But what Gödel formulates in the above could be formulated even more strongly. One could say that formal systems, in their use, are not static, but are constantly developing, and that the development of formal systems is precisely the procedure envisioned by Gödel that can actualize the development of greater precision and scope in the use of abstract concepts.

I see Gödel as one of the few thinkers (if not the only thinker) to bridge the gap between constructive and non-constructive modes of thought, and he brought that ability to think both constructively and non-constructively to his idea of formal systems, which is, as a consequence, highly subtle and nuanced.

Thus while Gödel might have agreed that the only precise idea of precision we possess is intrinsically finitistic, I think he would have also pointed out that any one precise idea of precision could be infinitely transcended by more precise ideas of precision -- that is to say, it would be possible to extrapolate a hierarchy of precision and rigor, with formal systems embodying a given level of precision and rigor at each stage of that hierarchy. 

In A Precise Idea of Precision I considered how the seven forms of vagueness identified and analyzed by J.L. Austin might be used to formulate seven forms of precision in contradistinction to each form of vagueness.

It is easy to see that precision and vagueness can coexist, and in any formulation of a given idea we are likely to find precise elements jumbled together with imprecise elements -- if one cared to do so, one could select from Austin's list of seven forms of vagueness and the parallel seven forms of precision in order to make this point. Moreover, a focus on any one aspect of precision is likely to come at the cost of the neglect of other forms of precision, unless one makes a conscious and dedicated effort to embody formal precision across the board. (As we all know, in quantum theory there is a direct trade-off of precision, such that precision in the measurement of the direction of a particle, for example, comes at the expense of precision of the measurement in the velocity of the particle, and this trade-off is precisely quantified by the Heisenberg Uncertainty Principle. It’s not always so neat.)

We see this imprecise intermingling of precision and imprecision even in the tradition of rigorous mathematics, which, I have previously observed, always involves a tension between intuition and formalization. Thus an axiom system renders our deductions rigorous, but at the cost of accepting axioms without proof. Rigorous mathematics demands precision, but it also demands the intuitions that underlie all mathematics and which make mathematics possible. Since mathematics is the oldest and most developed of all the sciences, the essential relationship to mathematical intuition has become obscured, and is even, on occasion, obfuscated, but the role of intuition cannot be entirely banished, no matter how devoutly this is desired.

There is a wonderful quote to this end from Hermann Broch:

“For positivism, intuition becomes a kind of secret and unpermitted love, while the positivistic moral, so thirsty for purity, anti-metaphysical, and anti-ontological, dictates that the visits to his mysterious lover are paid -- even though sterile -- more rigorously in incognito.”

That axioms are accepted without proof does not mean that they are wholly arbitrary. The formation of axioms is governed by unspoken conventions. That doesn't make them any less formal or any less mathematical, or deductions made therefrom any less rigorous, but it is as much as to acknowledge that the constitution of axioms is itself intuitive, i.e., one of the regions of contemporary mathematics that continues to employ mathematical intuition. Our intuitions may be refined and educated by previous formal experience, but we cannot reduce the formation of a formal system to a formal process. In other words, formal systems are not reflexively formal. 

In so far as a formal system is the only known embodiment of a precise idea of precision (according to Heyting), and in so far as a formal system is intrinsically finitistic (in both conception and execution), the only precise idea of precision we possess is intrinsically finitistic. This could be formulated as a syllogism, and this syllogism would seem to provide an epitaph upon infinitistic formal reasoning -- except that there is a problem here.

We are driven to the infinite in the pursuit of context, generality, formality, and extrapolation. With any step taken there is an immediate realization that another step can be taken; that this admits of no intrinsic limitation is the origin of mathematical induction as a technique of proof. Formal systems, too, must be placed in context, in intellectual context, and extrapolating the context of a formal system to an idealized generality is a process that converges on infinity. Even finitistic mathematical reasoning in finitistic formal systems implies the infinitistic context of this reasoning and these formal systems.

We have a theoretical context for conceptualizing the infinitistic context of our finite thoughts in Cantor’s work, though this has not been exploited as it might. Although set theory revolutionized mathematics in the twentieth century, the bulk of Cantor's work on transfinite numbers remains an outlier in mathematics research -- not unlike the way in which mathematicians have absorbed the formal lessons of Brouwer's intuitionism through Heyting while dispensing with Brouwer's mysticism. Just so, mathematicians have absorbed the lessons of Cantor's set theory through the axiomatizations of Zermelo (followed by the revisions of Fraenkel and Skolem) while largely dispensing with Cantor's transfinitism.

The infinitistic context of finite human life, thought, and reason is related to the problem of generalization, because generalization admits of the possibility of its infinite extrapolation. Can we conceptualize a pure generalization of a formal system or formal mathematical reasoning? Could there be any such thing as precise generality, i.e., a generality that would preserve the rigor of mathematical thought? And if we demand the extrapolation of our finitistic framework of formal thought, rather than arrive at a pure generality, would we not eventually and ultimately find ourselves in an exploration of the evolutionary psychology of mathematics, because of the history of human thought does not go back into the infinite past, but has a definite beginning in time?

These admittedly difficult questions point to an ambiguity that needs to be resolved that suggests a distinction between real (or empirical) and ideal formal thought -- eerily similar to Hilbert’s philosophy of mathematics. Perhaps it would be worthwhile to return to Hilbert at this juncture and to see what insights he has to offer in this vein. Is this a distinction -- between the real and the ideal -- that is useful today?  

Well, I’ve gone quite far afield without ever focusing on the central (and difficult) issue that I wanted to broach in relation to precise generality. But I sometimes find that, when I am pursuing a difficult and elusive idea, that I need to come at it from many different directions and by many different strategies, and by these indirections my direction find. So this inquiry into precisely generality is not at an end, but only at its beginning. Fate willing, I’ll return to this soon, either by way of an inquiry into the radical empiricism of Sextus Empiricus, or by way of the distinction between the formal and the general.