Oct 13, 2012
The Blow of the Epistemology of Mathematics: Turan’s and Posy’s Position between The Cartesian Doubt and Kant's Synthetic A Priori
Yogyakarta State University
Turan, H., 2004, elaborated that Descartes called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming that all beliefs derive from perception seems to rest on the presupposition that the Cartesian problem of doubt concerning mathematics is an instance of the problem of doubt concerning existence of substances.
Turan 1 argued that the problem is not whether we are counting actual objects or empty images but whether we are counting what we count correctly; he argued that Descartes's works is possible to expose that the proposition '2+3=5' and the argument 'I think, therefore I am,' were equally evident.
According to Turan 2, Descartes does not found his epistemology upon the evidence of mathematical propositions; and the doubt experiment does not seem to give positive results for mathematical operations.
According to Turan 3, consciousness of carrying out a mathematical proposition is immune to doubt; and statements of consciousness of mathematical or logical operations are instances of 'I think' and hence the argument 'I count, therefore I am' is equivalent to 'I think, therefore I am’.
Turan 4 indicated that if impugning the veridicality of mathematical propositions could not pose a difficulty for Descartes's epistemology which he thought to establish on consciousness of thinking alone, then he cannot be seen to avoid the question.
Turan 5 concluded that discarding mathematical propositions themselves on the grounds that they are not immune to doubt evoked by a powerful agent does not generate a substantial problem for Descartes provided that he believes that he can justify them by appeal to God's benevolence.
Turan, H., 2004, insisted that a relation between perception and mathematics is incontestable, however confining our thoughts to a context where the ontological presuppositions of un-philosophical reflection on perception are at stake; according to him, we must note the significance of perception with respect to the nature of existence that Descartes considers primarily for epistemological purposes. Turan 6 noted that Descartes seems to abandon the deceiving God argument for the demon assumption and this last hypothesis seems to call into doubt exclusively beliefs related to existence of an external world; therefore, it is possible to argue that Descartes gave up pursuing the question concerning the veracity of the mathematical judgments; and Descartes seems to endow the evil genius solely with the power of deceiving him in matters related to judgments on the existence of external things. Turan 7 found that Descartes always considered mathematical demonstrations among the most evident truths that human mind can attain, and referred to them as examples of objects which can be intuited clearly and distinctly; Descartes perceived that arithmetic and geometry alone are free from any taint of falsity or uncertainty. According to Descartes 8, mathematics is concerned with an object so pure and simple that they make no assumptions that experience might render uncertain; it consists in deducing conclusions by means of rational arguments.
Next, Turan, H., 2004, insisted that Descartes put on external existence of the objects in question; he considered both deduction and intuition as legitimate methods of acquiring knowledge. For Descartes 9, intuition is simple indubitable conception of a clear and attentive mind which proceeds solely from the light of reason and on that account more certain than deduction, but deduction is not epistemologically inferior to intuition for the attentive human mind. Descartes claimed that although mathematics makes extensive use of deduction. Descartes 10 does not say that deduction is the sole legitimate method of this domain and holds that intuition is as indispensable as deduction for the mathematical body of knowledge; and mathematical propositions had the same degree of certainty as the indubitable ontological argument cogito. Turan 11 indicated Descartes none the less always related the evidence of mathematical propositions to exactitude of mathematics which he thought to be deriving from the simplicity of their objects and hence to its ontological un problematical nature. For Descartes 12 mathematics is invariable with respect to ontological presuppositions, but once carried into the context of the doubt experiment it is seen that it bears crucial ontological implications that is it appears that mathematical objects and operations presuppose existence. Descartes then claimed that:
I perceive that I now exist, and remember that I have existed for some time; moreover, I have various thoughts which I can count; it is in these ways that I acquire the ideas of duration and number which I am then able to transfer to other things. As for all the other elements which make up the ideas of corporeal things, namely extension, shape, position and movement, these are not formally contained in me, since I am nothing but a thinking being; but since they are merely modes of a substance and I am a substance, it seems possible that they are contained in me eminently. 13
Further, Turan, H., 2004, insisted that functional and ontological dependence of number and other universals, renders cogito in which an instance of thought where both evidence and ontological certainty could be attained in a single step; epistemologically prior to mathematical propositions which may, it considered apart from the context of the doubt experiment and seen to embody evidence. According to Turan 14, 'I count, therefore I am' is epistemologically equivalent to 'I think, therefore I am'; both arguments are immune to doubt; however, the evil genius can indeed make me go wrong as I count my thoughts or the appearances, but cannot deceive me in the inference I draw there from the fact that I am counting is sufficient to prove that I exist regardless of whether or not I count or add or perform any mathematical operation erroneously. Turan 15 concluded that the ontological situation established by the Cartesian experiment of doubt brings in serious epistemological constraints; the experimenter discovers that any epistemological means he may want to employ for a further ontological move must necessarily be one available from the proper resources of the ontological situation he has confined himself to for epistemological purposes; in other words, the epistemological standards the experimenter must conform to are determined by the ontological setting of the doubt experiment. Turan 16 noted that hence the experimenter finds him self alone with things which we may call perceptions or thoughts, at a standpoint from where he attests to happenings of perceptions and thoughts and cannot know well how they are procured; while Descartes could therefore depend solely on the thought that he has perceptions or thoughts in his epistemological inquiry to establish a certainty which may not be affected by the arguments of the doubt experiment.
Posy, C., 1992, elaborated that prior to Kant, mathematics is about the empirical world, but it is special in one important way that necessary properties of the world are found through mathematical proofs; however to prove something is wrong, one must show only that the world could be different. In term of epistemological problem, Posy 17 notified that sciences are basically generalizations from experience, but this can provide only contingent, possible properties of the world that is it could have been otherwise. On the other hand, science simply predicts that the future will mirror the past; while mathematics is about the empirical world, but usually methods for deriving knowledge give contingent knowledge, not the necessity that pure mathematics gives us; in sum, Posy concluded that Kant wants necessary knowledge with empirical knowledge. Posy 18 then exposed Kant’s solves the problem in a couple of steps: first, that objects in the empirical world are appearances or phenomenon in which, by their nature, they have only the properties that we come to know of them from experiences; they are not things in themselves. Posy 19 found that Kant said we must become an idealist in which object's properties are only what is perceivable; there are no non-experienceable properties of objects. Second, Kant 20 suggested to build into our minds two forms of intuition and perception such that every perception we have is shaped by the forms of Space and Time; according to Kant, these are, in fact, parts of the mind, and not something the mind picks up from experience; and thus, empirical objects are necessarily spacio-temporal objects.
Next, Posy, C., 1992, indicated that, according to Kant, we come to know spacio-temporal properties in an a priori fashion; and in studying spacio-temporal properties, we are merely studying ourselves, and our perceptual abilities. According to Kant 21, mathematics is simply the science that studies the spacio-temporal properties of objects by studying the nature of space and time; and thus, mathematics is the studying of the abstract form of perception. In term of infinitary ideas that things is not subject to perception, Kant, as it was indicated by Posy, makes a distinction between empirical intuition that is the intuition from the senses which is always finite and pure intuition. Posy 22 indicated that the study of possibilities for empirical intuition where finite limits are not introduced in either direction; and mathematics doesn't deal with this. According to Kant 23 mathematics can allow the division of small intervals and the expansion of large intervals; this means we can discuss smaller and smaller quantities without introducing the smallest quantities e.g. if we want to prove an interval is divisible, we can do this by picking an interval; showing it is divisible; and abstracting from its actual size, and let it represent the notion of a perceivable interval.
Kant 24 claimed that pure mathematics, as synthetical cognition a priori, is only possible by referring to no other objects than those of the senses, in which, at the basis of their empirical intuition lies a pure intuition (of space and of time) which is a priori. Kant 25 claimed that this is possible, because the latter intuition is nothing but the mere form of sensibility, which precedes the actual appearance of the objects, in that it, in fact, makes them possible; and yet this faculty of intuiting a priori affects not the matter of the phenomenon. Kant 26 illustrated that in ordinary and necessary procedure of geometers, all proofs of the complete congruence of two given figures come ultimately to this that they may be made to coincide; which is evidently nothing else than a synthetical proposition resting upon immediate intuition, and this intuition must be pure, or given a priori, otherwise the proposition could not rank as apodictically certain, but would have empirical certainty only. Kant 27 further claimed that everywhere space has three dimensions, and that space cannot in any way have more, is based on the proposition that not more than three lines can intersect at right angles in one point. Kant 28 argued that drawing the line to infinity and representing the series of changes e.g. spaces travers by motion can only attach to intuition, then he concluded that the basis of mathematics actually are pure intuitions; while the transcendental deduction of the notions of space and of time explains, at the same time, the possibility of pure mathematics.
Because it would be absurd to base an analytical judgment on experience, as our concept suffices for the purpose without requiring any testimony from experience, Kant 29 concluded that Empirical judgments are always synthetical, e.g. “That body is extended” is a judgment established a priori, and not an empirical judgment. And also, for before appealing to experience, we already have all the conditions of the judgment in the concept, from which we have but to elicit the predicate according to the law of contradiction, and thereby to become conscious of the necessity of the judgment, Kant concluded that which experience could not even teach us. According to Kant 30, Mathematical judgments are all synthetical and he argued that this fact seems hitherto to have altogether escaped the observation of those who have analyzed human reason; it even seems directly opposed to all their conjectures, though incontestably certain, and most important in its consequences. Further he claimed that for as it was found that the conclusions of mathematicians all proceed according to the law of contradiction, men persuaded themselves that the fundamental principles were known from the same law. “This was a great mistake”, he said. He then delivered the reason that for a synthetical proposition can indeed be comprehended according to the law of contradiction, but only by presupposing another synthetical proposition from which it follows, but never in itself.
Similarly, Kant 31 argued that all principles of geometry are no less analytical. He illustrated that the proposition “a straight line is the shortest path between two points”, is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality. He claimed that the attribute of shortness is therefore altogether additional, and cannot be obtained by any analysis of the concept; and its visualization must come to aid us; and therefore, it alone makes the synthesis possible. Kant 32 confronted the previous geometers assumption which claimed that other mathematical principles are indeed actually analytical and depend on the law of contradiction. However, he strived to show that in the case of identical propositions, as a method of concatenation, and not as principles, e. g., a=a, the whole is equal to itself, or a + b > a, the whole is greater than its part. He then claimed that although they are recognized as valid from mere concepts, they are only admitted in mathematics, because they can be represented in some visual form. Posy 33 concluded that there are two consequences of Kantian view that no such thing as unapplied mathematics i.e. mathematics is, by nature, about the world; if it's not, it's just an abstract game; and that there is exactly one right mathematical theory of time, space, and motion.
1 Turan, H., 2004, The Cartesian Doubt Experiment and Mathematics,http://www.bu.edu
17Posy, C., 1992, Philosophy of Mathematics, http://www.cs.washington.edu/ homes/ gjb.doc/philmath.htm