The Varieties of Non-Constructive Experience
On Not Knowing How We Know What We Know and Related Puzzles
In a post of some years ago, Boundary Skirmishes in the Limits of Scientific Knowledge, I identified as one of the possible epistemic positions we can take in regard to the big bang the proposition, “There was something before the big bang, but we can’t know what it is.” Thinking about this later I realized that this is a spectacularly non-constructive claim, but not a non-constructive claim of existence or knowledge, but rather the non-constructive claim that something cannot be known (what could be called epistemically non-constructive, in contradistinction to the ontologically non-constructive), which also suggests the parallel possibility of a non-constructive claim that something does not or cannot exist (i.e., the ontologically non-constructive).
There are certain paradoxical aspects to asserting that something cannot be known. The most radical skeptical claim is that nothing at all can be known, which is a position sometimes identified as Pyrrhonic skepticism, though this form of skepticism is a perennial feature of philosophical thought. Francisco Sánchez, for example, defended this view in That Nothing is Known (Quod nihil scitur, 1581). What is it that we claim cannot be known? If we can identify this, then we have some small measure of knowledge of it. This makes it paradoxical to assert that something cannot be known.
In contradistinction to a non-constructive claim of that which cannot be known, a non-constructive epistemic claim would be to assert that one knows something without knowing how one knows the object of knowledge in question. Thus non-constructive epistemic claims are nothing other than what Polanyi called tacit knowledge, and is as familiar to us as recognizing a face without being able to explain how you recognized the face. There is nothing new in this except to recognize that it is a form of non-constructive experience. However, a distinction can still be made between asserting that one does not in fact know how one knows something, and asserting that one cannot know how one knows that which one does in a fact know to be the case. The former could be mere ignorance; the latter is a strong claim, and perhaps coincides with a claim of ineffability.
A non-constructive claim that something does not or cannot exist (i.e., the ontologically non-constructive) also puts one in the paradoxical situation of having to identify that which does not exist, which seemingly puts one in the position of acknowledging something to be true about the non-existent, granting it a kind of shadow existence. G. E. Moore made a distinction between existence and being to account for this problem, which “being” used to identify this shadowing form of existence, but the most famous philosophical doctrine to deal with this problem was Russell’s theory of descriptions, which analyzed propositions that seemed to name non-existent objects into propositions in which the seeming reference vanishes, and so also the apparent need to bestow “being” upon them while denying “existence” to them. In any case, there is a long philosophical history attached to naming non-existents like the round square or the wooden iron, and as a result conceptual techniques have been developed in order to handle ontologically non-constructive claims in the negative.
Russell’s favorite example was the present King of France, which he asserted was bald. Russell’s technique for doing away with this phantom was to employ quantification: there is one and only one thing, x, such that x is the King of France, x is alive to today, and x is bald. This circumlocution provides us with a proposition that has an unambiguous truth value, viz. false, because no such x exists that fulfills the stipulations of the proposition. But does Russell simply show that there is no present King of France, bald or otherwise, or that it is impossible that there should be a bald present King of France? This is not answered by the theory of descriptions insofar as I understand it.
Impossibility is much a stronger claim that mere non-existence, which latter can be settled by an exhaustive account of the world, and which could change at any time. If France returned to monarchical government, and the newly anointed king was bald, then Russell’s circumlocution would have a different truth value, viz. true. Impossibility asserts something beyond mere non-existence, though we could make a distinction between impossible at the moment, and impossible
Generally speaking, claims of impossibility are usually non-constructive claims, as the assertion of impossibility usually comes without an explanation, though it certainly is possible to assert the impossibility of something existing or the impossibility of possessing some knowledge and then explaining exactly how the putative existent or object of knowledge cannot be exhibited or constructed. But, depending upon what Kant meant by an object being exhibited in intuition, if an object is impossible, it could not be so exhibited. A non-existent object might arguably be exhibited intuition, as it is not difficult to imagine a present King of France who is bald, but an impossible object like a round square or a wooden iron cannot be the object of coherent intuition.
A non-constructive existence claim (in contrast to a non-constructive non-existence claim) is to assert the existence of something without providing a method by which the existent in question can be exhibited or constructed. Now, since I say, “exhibited or constructed” obviously this could be formulated either way exclusively, each of which has a distinctive meaning (or could be given a more distinctive meaning if more sharply defined). To formulate constructive or non-constructive thought in terms of exhibition (or lack thereof) is to recur to the classic Kantian formulation (Kant, as I have noted, was a proto-constructivist) in terms of being exhibited in intuition. This is probably better than a formulation in terms of construction, which usually (thought not always) means that one is talking about mathematics.
To formulate constructive or non-constructive thought in terms of construction, on the other hand, raises the question of what activities count as construction and what objects of thought can be understood as having been constructed. Of course, the same can be said of formulations in terms of exhibition — this raises the question of what conscious activity counts as exhibition in intuition, and what objects of thought can be understood as having been exhibited. For this, at least, we have the Kantian tradition to guide us. In the case of construction, most of our examples would be drawn from mathematics.
While logical empiricism (and logical atomism) spoke in terms of the objects of the world being logical constructions (and so could be said to have been constructed), we more typically think of mathematical objects as being constructed, thus the most notorious example of a non-constructive existence claim is that of the axiom of choice, which asserts the existence of a set composed of elements each taken out of a subset of of an infinite set in question, but without giving an explicit method for constructing this choice set. Part of the objection is related to the problem of infinitistic choice sets, which cannot be carried out in the empirical world and thus are ideal, and another part of the problem is simply the assertion of existence without further evidence. These two objections could be isolated from each other in order to further sharpen the idea.
Another interesting distinction could be made within constructive existence claims between claims that provide a finite method for determining the validity of the construction, and claims that do not have a finite method for determining the validity of some construction. Of course, as soon as we bring in any infinitistic element, constructivism that takes the form of finitism means that the idea will not be constructivistically acceptable to all constructivists. Whether or not infinitistic reasoning is the ultimate deal-breaker for all constructivism is a question that I have often pondered. One could argue that Brouwer’s rejection of the law of the excluded middle was motivated by the application of the law of infinite domains, and that impredicative formulations involve a infinite regression, so the the infinite is present in one way or another in reasoning proscribed by the various schools of constructivism.
These various schools of constructivism have forced us to acknowledge that constructivism is multifarious, even if it is motivated by a unified desire to avoid infinitistic reasoning. Carnap, further, in his Logical Syntax of Language, shows how a number of constructivist formulations can be ranged from weaker to stronger claims, again demonstrating the varieties of constructivism. But non-constructivist thought has not been explicitly formulated or defended as such, that is, as a distinctive form of reasoning that ought to be consciously and purposefully pursued, expanded, and extended, and so we do not have a body of logical work that shows us the distinctive nature of non-constructive thought.
As can be seen from the many distinctions we can make if we attempt to clarify non-constructive thought, there is a multiplicity of non-constructive forms of experience no less than constructive forms.
Earlier posts on Constructivism (and Non-constructivism):
- P or Not-P
- What is the Relationship between Constructive and Non-Constructive Mathematics?
- A Pop Culture Exposition of Constructivism
- Intuitively Clear Slippery Concepts
- Kantian Non-Constructivism
- Constructivism without Constructivism
- The Vacuous Identity Principle
- Permutations of Infinitistic Methods
- Methodological Differences
- Constructivist Watersheds
- Constructive Moments within Non-Constructive Thought
- The Principle of Vacuous Pragmatism
- Addendum on Precise Generality
- The Two Philosophies of Mathematics
- From Gödel to Historiography
- Gödel between Constructivism and Non-Constructivism
- The Natural History of Constructivism
- Cosmology: Constructive and Non-Constructive
- Saying, Showing, Constructing
- Arthur C. Clarke’s tertium non datur
- A Non-Constructive World