Elaborate Kant's theory of space and time. How does this theory enable him to explain how mathematical propositions can be both synthetic and a priori?

Kant’s Theory of Space and Time

Kant’s theory of space and time is central to his transcendental idealism. He argues that space and time are not empirical concepts derived from outer or inner sense, respectively. Instead, they are pure forms of intuition, meaning they are inherent structures of our minds that precede all experience. We do not discover space and time *in* the world; rather, we impose them *on* the world as conditions for perceiving it.

Space as the Form of Outer Sense

Kant posits that all our experiences of external objects are necessarily structured by space. We cannot conceive of an object existing outside of spatial relations. Space is the form through which we perceive external sensations as being arranged in locations and distances. It is not a property of the objects themselves, but a feature of our way of representing them.

Time as the Form of Inner Sense

Similarly, time is the form of our inner sense. All our experiences, whether external or internal, are ordered in time. We cannot experience anything without placing it within a temporal sequence. Time is not an objective reality flowing independently of our minds, but the way we order our perceptions and thoughts.

Synthetic *A Priori* Judgments in Mathematics

Kant’s theory of space and time provides the foundation for his explanation of how mathematical propositions can be both synthetic and *a priori*. To understand this, we must first define these terms:

• Analytic Judgments: These are judgments where the predicate is contained within the concept of the subject. They are true by definition and do not expand our knowledge (e.g., “All bachelors are unmarried”).
• Synthetic Judgments: These are judgments where the predicate adds something new to the concept of the subject, expanding our knowledge (e.g., “All swans are white”).
• A Priori Judgments: These are judgments whose truth can be known independently of experience. They are necessarily true and universal.
• A Posteriori Judgments: These are judgments whose truth can only be known through experience. They are contingent and particular.

The Problem and Kant’s Solution

Traditionally, mathematical judgments were considered either analytic (and therefore trivial) or *a posteriori* (and therefore not necessarily true). Kant argued that mathematical judgments, such as “7 + 5 = 12,” are neither. They are not analytic because the concept of ‘12’ is not contained within the concepts of ‘7’ and ‘5’. They are not *a posteriori* because their truth is certain and universal, not dependent on any particular experience.

Instead, Kant argues that mathematical judgments are synthetic *a priori*. They expand our knowledge (synthetic) but are known independently of experience ( *a priori*). This is possible because space and time, as *a priori* forms of intuition, provide the framework within which mathematical operations take place. For example, the concept of addition is grounded in our intuition of time as a sequence. The certainty of mathematical truths stems from the certainty of the forms of intuition themselves.

Consider geometry. Kant argues that geometrical propositions, like “A straight line is the shortest distance between two points,” are synthetic *a priori*. The concept of ‘shortest distance’ is not contained within the concept of a ‘straight line’; it adds something new. Yet, we know this proposition is true not through empirical observation, but through our *a priori* intuition of space.

Judgement Type	Definition	Example
Analytic	Predicate contained in subject	All triangles have three sides
Synthetic	Predicate adds to subject	All triangles have angles summing to 180 degrees
A Priori	Known independently of experience	All objects are extended in space
A Posteriori	Known through experience	This table is brown