by Andrew Nightingale, November 14th, 2559

CC: Dr. Khajornsak Buaraphan, Dr. Parames Laosinchai, Dr. Patchayapon Yasri

The problem with Heideggers “enframing” attempt—that science enframes nature and in any frame there the phenomena are still vague—is that certain kinds of vagaries do not entail paradigm shift. “Though discovering life on the moon would today be destructive of existing paradigms (these tell us things about the moon that seem incompatible with life’s existence there), discovering life in some less well-known part of the galaxy would not.” (Kuhn 1970, p 95) However, some vagueness does warrant paradigm shift, because “Ambiguity [between terms and the world], … turns out to be an essential companion of change.” (Feyerabend 1999, p 39)

Precision, on the other hand, is not an argument in favor of a theory because “In fact, so general and close is the relation between qualitative paradigm and quantitative law that, since Galileo, such laws have often been correctly guessed with the aid of a paradigm years before apparatus could be designed for their experimental determination.” (Kuhn 1970, p 29) So that the measurements are predicted with intense precision, and then the experiment carried out is an elaborate, highly overdetermined one that has only one possible interpretation within the paradigm.

Vagueness is apparent to the naked eye, but it is traditionally opposed to what can be grasped rationally. “In general, Leibniz had followed the other great rationalists in interpreting perception as a confused form of thinking. Like Descartes, he had treated the deliverances of the senses as sometimes clear but never distinct.”(Walsh; Edwards Ed. 1972, p 307) However, vagueness is a clear and distinct concept, and it seems that it also is in complete agreement with the “deliverances of the senses.” Thus, in the sense of mathematics that Whewell and others held, vagueness is a truly mathematical one, that is,

“…in mathematics there was no difference between objective reality and subjective knowledge; the human mind was completely in tune with external fact.” (Richards 1980, p 362) Rational thought and empirical observation are brought together into one concept: vagueness. This old idea of mathematical truth has changed drastically now. With Godel’s theorems, it became clear that an absolutist (that is mathematics is absolutely true and unchanging) view became untenable. One stronghold of the old sense in which mathematics is true (Whewell’s) can be found in the mathematician Brouwer’s intuitionism. According to Brouwer (and Kant before him), the experience of time is accessed to fill the empty formalisms of mathematics, giving it meaning and truth. Vagueness is another source of mathematical truth. It may be that vagueness between two things is present in Brouwer’s intuition of a “twoity,” the beginning of intuitionist arithmetic.

What do I mean by vagueness? The ancient representation of vagueness is the problem of the heap of sand. When you have a heap of sand, you have a relatively safe inference that if you take one grain from a heap, then you will still have a heap. As the story goes, eventually taking grains of sand will show this if, then statement to be faulty because you will no longer have a heap of sand. Why does classical “if, then” fail us here? There is an analogue between the heap of sand example and with the calculation of a real number according to a rule. Also, this question gains importance when reflecting that “Logical consequence [the if, then] is the central concept in logic. The aim of logic is to clarify what follows from what. – Stephen Read, Thinking about Logic [99]” (As quoted in Beall, Restall 2006, Kindle Edition) According to Beall and Restall, logical consequence can be clarified in more than one way, giving rise to more than one equally valid (if applied in different situations) formulation of logical consequence. “We must reconcile ourselves to the fact that every precise definition of [logical consequence] will show arbitrary features to a greater or less degree.” (Tarski as quoted in Beall, Restall 2006)” Additionally probability theory is not a solution to the vagueness of logical consequence, because

…probability theory might provide a canon for evaluating degrees of belief, … Nonetheless, probability theory cannot be a complete answer here, for we also make assertions and denials (and hypotheses and many other things besides), and these may also be evaluated for coherence, using the norms of deductive logic. In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion… (Beall Restall 2006, Kindle Edition)

The solution to the vagueness of logical consequence, rather, lies in logical pluralism. Logical consequence brings true conditions to their true conclusions, but logical consequence itself is conditioned, and ultimately forms the structure of what can be intelligibly conditioned. Since phenomena are inherently vague, and logical consequence is vague until arbitrarily made precise, there is no clear difference between form and substance, ideas and things.

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Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.

Edwards, P., & Walsh, W. H. (1972). The encyclopedia of philosophy (2nd ed., Vol. 4). New York: Macmillan.

Feyerabend, P., & Terpstra, B. (1999). Conquest of abundance: A tale of abstraction versus the richness of being. Chicago: University of Chicago Press.

Kuhn, T. S. (1970). The structure of scientific revolutions. Chicago: University of Chicago Press.

Richards, J. L. (1980). The art and the science of British algebra: A study in the perception of mathematical truth. Historia Mathematica, 7(3), 343-365. doi:10.1016/0315-0860(80)90028-2