Table of Contents

1. Immanuel Kant
2. Answering Hume
3. Kant’s Ingenious Idea
4. From Article on Kant Stanford Encyclopedia of Philosophy
5. Synthetic a priori Judgments
6. Analytic and Synthetic Judgments
7. Refutation of Kant’s Theory
8. Synthetic a priori Propositions of Mathematics
9. Synthetic a priori Propositions of Natural Science
10. Kant’s Critique of the Ontological Argument
11. Conclusion

Immanuel Kant

Answering Hume

• Descartes, Spinoza, and Leibniz claimed that certain facts beyond the truths of mathematics can be known a priori, for example:
  • Descartes’ principle that what is clearly and distinctly perceived is true
  • Spinoza’s axioms
  • Leibniz’ Principle of Sufficient Reason.
• Hume argued against the Rationalists that no matter of fact can be known a priori:
  • A proposition can be known a priori only if its negation implies a contradiction.
  • The negation of a matter of fact does not imply a contradiction.
  • Therefore, a matter of fact cannot be known a priori.
• Kant agreed with Hume that the arguments of the Rationalists failed, e.g. the Ontological Argument. But he also thought that Hume was mistaken.
• Kant set himself the goal of answering Hume, explaining how certain facts about reality could be known a priori.  As he put it, “How are synthetic a priori judgments possible?”

Kant’s Ingenious Idea

• A tomato is a thing-in-itself, or noumenon, about which nothing is known except its existence. In perception, the human mind interacts with a noumenal tomato, imposing an a priori conceptual structure on the tomato, resulting in the perception of a phenomenal tomato, existing in time and space, having primary and secondary qualities, and causally interacting with other phenomenal objects.
• Since the phenomenal tomato results from the mind’s structuring the noumenal tomato, Kant argued that certain facts about the phenomenal tomato can be derived a priori from how the structuring works.  The Critique of Pure Reason is devoted to deriving these facts.

From Article on Kant
Stanford Encyclopedia of Philosophy

“Kant’s revolutionary position in the Critique is that we can have a priori knowledge about the general structure of the sensible world because it is not entirely independent of the human mind. The sensible world, or the world of appearances, is constructed by the human mind from a combination of sensory matter that we receive passively and a priori forms that are supplied by our cognitive faculties. We can have a priori knowledge only about aspects of the sensible world that reflect the a priori forms supplied by our cognitive faculties. In Kant’s words, “we can cognize of things a priori only what we ourselves have put into them.” So according to the Critique, a priori knowledge is possible only if and to the extent that the sensible world itself depends on the way the human mind structures its experience.”

Synthetic a priori Judgments

• Kant’s objective was to explain how “synthetic a priori judgments” are possible.  He defined “synthetic” and “analytic” as follows:
• A statement is analytic if the concept of the predicate is contained in the concept of the subject. A statement is synthetic otherwise.
• Prolegomena to Any Future Metaphysics (Preamble)
  • “Analytical judgments express nothing in the predicate but what has been already actually thought in the concept of the subject, though not so distinctly or with the same (full) consciousness.”
• Critique of Pure Reason (A6–7)
  • “In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic.”

Analytic and Synthetic Judgments

• Kant’s concepts of analytic and synthetic differed from Hume’s relations of ideas and matters of fact and Leibniz’ truths of reason and truths of fact.
• For Hume and Leibniz, a mathematical truth is a relation of ideas / truth of reason, since its negation implies a contradiction.
• But for Kant mathematical truths are synthetic, not analytic.
  • “It might at first be thought that the proposition 7 + 5 = 12 is a mere analytical judgment, following from the concept of the sum of seven and five, according to the law of contradiction. But on closer examination it appears that the concept of the sum of 7 + 5 contains merely their union in a single number, without its being at all thought what the particular number is that unites them. The concept of twelve is by no means thought by merely thinking of the combination of seven and five; and analyse this possible sum as we may, we shall not discover twelve in the concept. We must go beyond these concepts, by calling to our aid some concrete image (Anschauung), i.e., either our five fingers, or five points (as Segner has it in his Arithmetic), and we must add successively the units of the five, given in some concrete image (Anschauung), to the concept of seven. Hence our concept is really amplified by the proposition 7 + 5 = 12, and we add to the first a second, not thought in it. Arithmetical judgments are therefore synthetical, and the more plainly according as we take larger numbers.” (Prolegomena, Preamble)
• So for Hume and Leibniz 7 + 5 = 12 is a relation of ideas, or truth of reason, because its negation implies a contradiction.  For Kant the equation is a synthetic judgment because the concept of 7 + 5 does not contain the concept of 12. Thus,  7 + 5 = 12 is both synthetic and a relation of ideas.

Refutation of Kant’s Theory

• In the Introduction of the Critique, Kant says that the problem of pure reason is to answer the question: “How are synthetical judgements a priori possible?”  Since it relates to the “sciences which contain theoretical knowledge a priori of objects,” the question divides into:
  • How is pure mathematical science possible?
  • How is pure natural science possible?
• That is,
  • How are the synthetic a priori propositions of mathematics possible? 
  • How are the synthetic a priori propositions of natural science possible? 
• Kant derives answers to these questions from how the mind structures things-in-themselves. The problem is that, for the first question, there is a far simpler answer and, for the second, Kant’s theory explains things that are false.

Synthetic a priori Propositions of Mathematics

• How are the synthetic a priori propositions of mathematics possible? 
• Grant that mathematical truths are synthetic in Kant’s sense, that their subjects don’t contain their predicates. Kant’s question is: how can mathematical truths be known a priori?  Hume has already answered the question, however: mathematical truths are relations of ideas and therefore can be proven a priori.
  • Relations of ideas “include every affirmation which is either intuitively or demonstratively certain.” (EHU 20)
  • Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe” (EHU 20)
• For example, the proposition that the product of two odd integers is odd is known a priori because it’s proved a priori:
  • Assume that n and m are odd integers; we want to prove that nm is odd.  Since n is odd there’s an integer r such that n = 2r + 1. And since m is odd there’s an integer s such that m = 2s +1.  Therefore nm = (2r +1)(2s + 1) = 4rs + 2s + 2r + 1 = 2(2rs + s + r) + 1. So there’s an integer, 2rs + s + r, such that nm equals twice that integer plus 1.  Therefore nm is odd.

Synthetic a priori Propositions of Natural Science

• How are the synthetic a priori propositions of natural science possible? 
• Kant tried to explain how the following are synthetic a priori
  • “Every event is determined by a cause according to constant laws” (Prolegomena)
  • “With all changes of corporeal nature, the quantity of the matter remains, on the whole, the same, unincreased and undiminished.” (Metaphysical Foundations of Natural Science)
• The problem is you can’t explain what’s false
  • The predictions of Quantum Mechanics are probabilities  For example, Quantum Mechanics predicts that the probability is 1/2 that a silver atom passing through a Stern-Gerlach magnet is deflected up rather than down. The path of  the atom is a matter of chance and thus not determined by the laws of nature.
  • Matter is not conserved. Mass is annihilated when an electron and positron collide.  Mass is produced when a fast gamma ray passing by a heavy nucleus transforms into an electron-positron pair.

Kant’s Critique of the Ontological Argument

• Kant in the Critique
  • “Being is evidently not a real predicate, that is, a conception of something which is added to the conception of some other thing. …. Now, if I take the subject (God) with all its predicates (omnipotence being one), and say: God is, or, There is a God, I add no new predicate to the conception of God, I merely posit or affirm the existence of the subject with all its predicates—I posit the object in relation to my conception.”
• Red vs Exist
  • “Tomatoes are red” means “if a thing is a tomato, it’s red” and ascribes a property to tomatoes.
  • “Tomatoes exist” doesn’t mean “if a thing is a tomato, it exists,” a logical truth, and doesn’t ascribe a property to tomatoes. It means, rather, “some things are tomatoes” and asserts existence.

Conclusion

• From the Britannica
  • “Kant was one of the foremost thinkers of the Enlightenment and arguably one of the greatest philosophers of all time.” (britannica.com/biography/Immanuel-Kant)
• My Comments
  • Kant’s criticism of the Ontological Argument, that “existence is not a predicate”, is on target.
  • 150 years before it was confirmed, Kant proposed that some nebulae are galaxies.
  • Kant’s idea that the mind imposes a conceptual structure on noumenal objects was ingenious. But ….
    • Hume had already explained how the synthetic a priori propositions of mathematics are possible.
    • Kant’s theory of the mental structuring the noumenal manages to explain why some false propositions of physics are true.