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Kant reflects upon Mathematical Judgements in his work entitled “Prolegomona:
“Judgements of experience are always synthetic…..Mathematical judgements are all synthetic….It might at first be thought that the proposition 7+5=12 is a mere analytic judgment, following from the concept of the sum of seven and five, according to the principle of contradiction. But on closer examination it appears that the concept of 7+5 contains merely their union in a single number, without its at all being thought what the particular number is that unites them…..We must go beyond these concepts by calling to our aid some intuition corresponding to one of them, i.e. either our five fingers or five points: and we must add successively the units of the five given in the intuition to the concept of seven. Hence our concept is really amplified by the proposition 7+5=12 and we add to the first concept a second one not thought in it.”(267-268)
It is not absolutely clear in the above reflection how Kant is conceiving of the intuition that we need to use in the activity of counting or adding five units to an existing seven. What is clear, however, is that there is an experience of counting which follows the rule “n+1”, five times, before the process comes to an end. Kant also claims that the mathematical proposition 7+5=12 is a synthetic a priori proposition because it is what it is in virtue of a necessity that is not merely true but justifiably true in a way that is independent of experience. The Prolegomena further differentiates a priori cognitions:
“The essential distinguishing feature of pure mathematical cognition among all other a priori cognitions is that it cannot at all proceed from concepts, but only by means of the construction of concepts.”(272)
At first glance this looks as if it is being claimed that the cognitive process in question is being created by the subsumption of sensory/schematic data(pure intuition) under the category of quantity, but Kant is very clear that it is reason and the principle of contradiction that is regulating mathematical activity. We are, nevertheless, dealing with intuitive judgements and their relation to the a priori intuitions of space and time. Geometry, it is argued is founded upon the pure intuition of space, and arithmetic the pure intuition of time. Kant’s account of space maintains that it by necessity cannot have more than three dimensions, and the “proof” for this is given by the synthetic a priori truth of geometry that not more than three lines can intersect at any one given point. This of course also depends on the condition that the space of geometers is the space that is given to human beings in the form of a sensuous intuition. Kant insists that our spatial cognition is both empirically real and transcendentally ideal. This form of idealism is critical, relating necessarily to our cognitive power of sensibility. Space and time, on this account, are modes of representations.
Time, Aristotle insisted, is the measurement of motion in terms of before and after. This is to some extent accepted by Kant, who elaborates upon this essence-specifying definition by situating it in relation to the categorial powers of the understanding, namely quantity and relation. The schemata related to our consciousness of time are, permanence, succession and coexistence. These schemata, Kant argues, contain the rules for all the possible temporal relations of appearances. These rules, Kant argues, regulate all possible temporal positions of objects and they are given their unity by the faculty of sensibility and the unity of apperception which in turn schematises its material via the categories of understanding/judgement(quantity, relation).
Kant elaborates upon this position in his first critique:
“Time is nothing but the form of inner sense, that is, of the intuition of ourselves and of our inner state. It is the immediate condition of inner appearances(of our souls) and thereby the mediate condition of outer appearances_(B49-51)
Kant is attempting to outline here the intuitive a priori sources of knowledge, which lie at the foundations of many of our sciences, including geometry and arithmetic. The science of motion, for example, presupposes that both of these a priori forms of intuition, which for Kant are both empirically real and transcendentally ideal, do not relate to the things-in-themselves that underlie appearances. They are nevertheless, not creatures of the imagination, but rather objective features of our experience. With such remarks, Kant is venturing deep into the territory of the challenge to humanity and philosophy handed down by the Delphic oracle, to “know thyself!”. The above remarks are in the name of theoretical pure reason, but there are also sciences that both Aristotle and Kant categorised as practical and productive that are concerned with the reasons and totality of conditions related to action rather than the justification of true beliefs. Time, in relation to Action and Art is more concerned with those forms of life that are ends-in-themselves(the good-in-itself) rather than those instrumental forms of life that relate means to ends and use practical reason instrumentally to calculate the consequences of actions. Life in a typical community embraces both the means and the ends of all forms of cognitive and normative activity. In our multi-purpose communities the oracular words of Ecclesiastes are especially appropriate:
“There is a time for every purpose under heaven”
This proclamation indicates the universal importance of temporality in our instrumental and categorical dealings with reality. History is one scientific discipline that deals with these purposes, describing and explaining them, but it is a discipline that crosses the boundaries of scientific domains, including as it does theoretical activity(looking for the causal relations between events) practical activity(all forms of activity instrumental and categorical that aim at the various forms of the good) and productive activity of various kinds. The principles we will find lying behind historical writings, therefore, will be theoretical/logical)principles of noncontradiction and sufficient reason), practical principles (freedom and the categorical imperative) and productiveprinciples(e.g. a time for every purpose under heaven).
How, then, does Wittgenstein’s account of mathematics fit in with the accounts we find in Aristotelian Hylomorphic philosophy or Kantian critical philosophy? Firstly we ought to remind ourselves of the fact that Wittgenstein viewed the Kantian project as very similar to his own, but it is not immediately obvious that were Kant to have the opportunity to pass judgement upon Wittgenstein’s work, the positive complement would have been returned. For Wittgenstein, the principle of sufficient reason is used to give an account of the essence of something and this is best achieved via grammatical investigations.
Wittgensteins reflections on the nature of arithmetic are clearly in the spirit of Kant. He points out, for example, that when you ask the following question of a child:
“I have in my right hand two apples and I have another two apples in my left hand. How many apples do I have altogether?”
If the child, in response to this question answered:
“How do I know, you may drop one apple and/or inherit an orchard of apples.”
Wittgenstein’s response to such an example would be to claim that this child does not understand the nature of mathematical activity. He does not understand, that is, that empirical experience cannot enter into the above imagined mathematical example, in the way suggested by the child. What we are dealing with here, instead, is a rigid rule-based activity which is aiming at a proposition composed of constructed concepts that are not determined by reality but are rather normative, and play an important role in determining judgements about numerical reality. Of course in the empirical world I could drop one apple, and/or be told that I have inherited an orchard but these experiential possibilities are irrelevant to the above mathematical exercise. The rigidity of the operation “add two” makes the outcome inevitable and everyone must agree to the proposition that results, or alternatively admit to attempting to introduce another arithmetical system no one is familiar with into the discussion. Given that we are dealing with the arithmetical system we have all learned at school and given the rigidity of the activity of counting, 1,2,3,4, there can be no other outcome than arriving at 4, and thus confirming the validity of the proposition 2+2=4.
For Wittgenstein it is the role or use of counting in our lives that serves to define its essence. Counting, in turn, is a necessary condition of the measurement of motion in our world, and this, in turn, for Aristotle is part of his account of the essence of time, namely “the measurement of motion in terms of before and after”. The relation of the clock and the calendar to counting the seconds, minutes, hours, days, months, and years of our life is obvious, both to everyman and to the philosopher. Time itself, can also be used in other sciences, such as History, in order to generate historical knowledge about the events, actors, communities, nations, and empires that form the “material” of historical judgements and understanding. Wittgenstein, then asks questions about how we learn to count, learn to tell the time and measure time, and learn about History. What we do, he argues, in our various forms of life determines the ultimate justifications of counting, time, and History.
History, as we have indicated, is composed of a number of forms of life, and uses of language, that stretch across the boundaries of theoretical, practical and productive knowledge. In these knowledge-contexts there will be synthetic a priori judgements such as “every event has a cause”, “every action has a reason”, “every purpose has its time”, and these judgments are all normative judgments of various kinds. Those who, in the name of the above remarks, embark on an explorative search to discover an event without a cause, an acton without a reason or a purpose that is not temporally embedded is misunderstanding the transcendentally ideal nature of the above normative judgments. Historical judgements also presuppose temporal judgments which in turn presuppose arithmetical judgments requiring the activity of counting. The activities, forms of life and uses of language and digits that constitute historical judgments and understanding for Kant, would be of several different kinds, including theoretical judgments about the causes of events and practical judgments involving the freedom of agents to choose particular courses of action.
Wittgenstein’s later work was both a response to his earlier work, as well as to the analytical Philosophy of his time. The concept of “language-games”, and what appeared to be their conventional nature took precedence over the more natural concept of “forms of life”, which in fact served to emphasise two important aspects of the later philosophy. Firstly, the reference to the importance of the way in which we learn these language-games reminds us of an organic view of language in which actualisation or development occurs over learning phases. Secondly, the conventional is, therefore, not opposed to human nature but rather reflects deep social needs.
We find in Wittgenstein’s “Remarks on the Foundations of Mathematics”, the following Kantian reflection:
“73. I could also have said: it is not the property of an object that is ever “essential”, but rather the mark of a concept”.(23a)
“Conventional” as an idea has often been opposed to the idea of what is “essential” because it is associated with what is subject to change at will, i.e. it is associated with what is transitory or arbitrary, rather than what remains the same throughout change. The problem with this form of reasoning is that once a community accepts a linguistic practice over a long period of time, change does not come easily: the uses of words and the forms of life in which they are embedded remain constant over long stretches of time. Change, when it does occur, does so for good reasons. On this line of reasoning, the “conventional” ought rather to be associated with some kind of necessity. For Wittgenstein, “what we do” is the synthetic aspect which we appeal to in any process of justification, but such a process also seeks to justify the necessity associated with customary practices that have endured over time. Such a process obviously has historical aspects.
logic, Wittgenstein argues, is constituted by so-called laws of thought which are not situated in a metaphysical Platonic realm of forms but rather show us what is essential in our thinking and reasoning activities. The “material” of thought can be intuitive but it can also be concepts that become part of reasoning processes that use “principles” or “essence specifying definitions” to establish the essential features of human nature(e.g. “rational animal capable of discourse”). Mathematical thought and reasoning uses the “material” of numbers, points, lines, and shapes and these elements are used in the mathematical process of the construction of concepts. Wittgenstein argues that the mathematician is:
“an inventor and not a discoverer”(47e)
This entails that mathematics, like language rests on a foundation of grammatical propositions and the acceptance of these propositions depends in turn upon the practice of accepting the necessity of following rules. The rule “n+1”, for example, could be changed if there was a good reason to do so, but this has not happened thus far in relation to our customary practice of counting in accordance with the above rule. Wittgenstein claims in several places, and not just in relation to the activities associated with mathematics, that we should seek to describe rather than explain these activities. For Wittgenstein describing “what we do” is a form of explanation and this is important for the purposes of preventing the argument that we can find ourselves caught in the trap of the infinite regress that might be generated by repeatedly asking for justifications for the justification. My spade is turned, it is argued, and we can go no deeper into the matter. Describing how we learned to use numbers, points, lines, and shapes is a part of a grammatical investigation. Such investigations are likened unto anthropological investigations.
We have pointed out previously that a Greek spirit dwells in the idea of “forms of life”: a spirit which assumes that life has an ultimate value as an end-in-itself. This is expressed thus:
“The limits of empiricism”(Do we live because it is practical to live? Do we think because it s practical?)” (171e Remarks)
Thinking, is therefore, also an end-in-itself in which conceptual material is formed both by the categories of the understanding and the principles of reason(noncontradiction, sufficient reason) and this form of activity is best exhibited in the human form of life that Aristotle characterises in terms of a “rational animal capable of discourse”.
Similarly, Wittgenstein argues that we do not count just because it is practical. The activity of thinking is something we humans do, as part of a larger context of activities, that relate to Time and the understanding of History. Activities can be conditions of possibility of the existence of other activities. He also argues that mathematical activities have a normative character and are part of a network of norms and rules that are integrated into many different kinds of activities necessary for humans to live together and flourish.