12. The Twelfth Lecture: The Form of Time and Space

In this lecture, we explore time and space, focusing on time as the a priori form of inner intuition that is instrumental in comprehending a priori synthetic judgments. What students typically come to realize as a result of this is far deeper and more significant than merely "placing objects into time." They come to understand, on a much deeper level, that time as an intuitive form resides within their subjectivity, shaping all perception and experience. This realization further illuminates how space, as the a priori form of outer intuition, becomes comprehensible—such as when avoiding stumbling over a chair or any other object while walking toward it. Passing the chair already demonstrates the use of these inherent abilities, synthesizing sensory data through time and space, even before one consciously notices them.

In other words: what we think we do is not what we actually do. We often believe, in a Newtonian manner, that we see objects, decide, and act, whereas it is far more accurate to conceptualize this as us only consciously reflecting on what we have already intuitively done.

Remember 5+7? Let us now explore more concretely what we did to reach the answer. Namely, we sequenced the counting of numbers within time, thereby gaining the ability to arrive at the solution. When it comes to geometry (or natural science), we follow a similar process, using the intuition of “space” to make a priori judgments. However, the intuition of space is only possible through the form of time. Kant states: “Time is an a priori condition of all appearance in general, and indeed the immediate condition of the inner intuition (of our souls), and thereby also the mediate condition of outer appearances.”

Kant was thus convinced that the human mind possesses a priori “built-in” capabilities, such as categories of understanding and forms of intuition, which are required to generate a priori synthetic judgments. In this sense, Kant sought to understand the exact mechanics, asking: “How do we attain such concepts [that inhabit the human mind] a priori?” The deceptively simple arithmetic of combining two numbers into one is an excellent example of how analytic judgment falls short. There is nothing within pure logic that allows us to deduce the correct answer, as numbers themselves are mere signifiers. We need to comprehend what they signify, which necessitates a sense of temporality and time.

Objects in themselves contain no time; however, objects that we can know (phenomenal objects) can be “placed in time” through our subjective sense. Kant states: “Time is therefore merely a subjective condition of our (human) intuition (which is always sensible, i.e., insofar as we are affected by objects), and in itself, outside the subject, is nothing.”

Thus, we can conclude that we could never experience an object without being able to place it “in time” within a framework through which we experience and understand objects. In this sense, our ability to experience the world (as phenomenal reality) is intrinsically connected to our a priori intuition of time (and space) as the subjective conditions of consciousness.

However, there is one important caveat: we cannot cognize or intuit time (or space) without any objects, as pure actuality in itself. As Kant stated, “In itself, /…/ [time] is nothing.” Conversely, when there is an object, we can (and even must) apply our subjective a priori intuition of time to generate an understanding of it. This understanding occurs through a representation in our mind, aided by our imagination.

Additionally, when we consider things in themselves (which we can never experience, as they belong to the realm of noumena), we cannot perceive them "in time" through reason. For example, we can never state that our primordial subjectivity ("S0" according to the model of Evolutionary Truth, or "transcendental self" in Kantian terms) can be observed in time as an object. This is because we cannot know its substance, location, qualities, duration, or other properties.

In the simplest terms, for objects we can comprehend (the phenomenal realm of appearances in our mind), our “built-in” intuitions of time (and space) must be employed to make sense of them and structure our reality. However, for objects we cannot comprehend (the noumenal realm of things-in-themselves), these cannot be connected “into time” (or space) within our subjective senses, nor do they contain any property of time or space in themselves.


Mathematics as Thinking in Time and Space

Now, when we return to “7 + 5 = 12,” we can see that, per Kant, mathematics, as our mind intuiting a synthetic judgment a priori, becomes possible because we can use our “built-in” inherent forms of intuition, especially the form of time. As he says, to perform mathematics, 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.” This is something we cannot imagine in our minds without the form of time. Let us be even more precise.

When we analyze solving this (rather simplistic) arithmetic problem, what happens in our mind is the following: we give our mind the “command” to solve this mathematical problem (as we have the freedom to do that). Then, our mind performs an operation, which consists of generating a succession of 5 units that it adds to a succession of 7 units; after that, it reaches a concept of 12 units. This conceptualization presupposes time (and imagination, which may also be related to objects in space, i.e., our fingers or points) to aid in the process.

The mind creates an idea of succession using the a priori forms of time and space, which enable us to reach an understanding within the mind without sensory input from the outer world. Our reason then utilizes the ideas and concepts generated within this framework (with the help of time) to carry out further operations and develop new concepts (i.e., solve more complicated mathematical problems).

The key idea here is that our a priori form of time does not solve mathematical problems independently; instead, it provides the necessary framework for our minds to perform mathematical operations. Our reason then utilizes the ideas and concepts generated within this framework to carry out further operations and develop new concepts.

If one says, for example, that one cannot perform complicated mathematical problems in the same way, it does not refute Kant’s description of the operations of the mind in the slightest. Regardless of the person's abilities, attempting to solve any mathematical problem demonstrates the involvement of a priori intuitions of time (and space) in our cognitive processes. Thus, this supports Kant's overall approach to understanding the role of a priori forms of intuition in the human mind.

In the simplest terms, mathematics is an a priori judgment made possible using our built-in intuition of time (and space). Regardless of our ability to solve a mathematical problem, our mind can conceptualize it within the context of time (and space).