A Critique of Immanuel Kant’s Concept of Synthetic A Priori Knowledge

The concept of synthetic a priori knowledge, introduced by Immanuel Kant, combines two philosophical concepts: synthetic and a priori. Synthetic knowledge refers to informative statements where the predicate adds new information not contained in the subject and a priori refers to knowledge attainable without the need to experience. In his 1781 book, Critique of Pure Reason, Kant controversially argued that some truths are both synthetic and a priori—these are foundational to human cognition, structuring how we perceive reality. Immanuel Kant used mathematics, particularly arithmetic and geometry, as a central example to illustrate his concept of synthetic a priori knowledge. He argued that mathematical truths are both universally necessary and informative, making them a perfect demonstration of how synthetic a priori judgments structure human understanding. However, Kant’s concept of synthetic a priori knowledge has been subject to significant criticism and debate. Critics argue that mathematics and other supposed examples of synthetic a priori knowledge may not fit his framework, and some even question whether synthetic a priori knowledge is possible at all. In agreement with the aforementioned stance, this essay will scrutinise Kant’s example of mathematics to demonstrate how it does not belong to the category of synthetic a priori.

Kant’s claim that mathematical truths like “7 + 5 = 12” or “the sum of angles in a triangle is 180 degrees” are synthetic a priori overlooks the foundational role of human-defined language and conventions in mathematics. The symbols, expressions, and definitions are not inherent truths of nature but arbitrary constructs governed by agreed-upon rules. The statement “7 + 5 = 12” is tautologically true only because we have predefined the symbols and operations of arithmetic operating within a base-10 system, a convention that humans devised for practical purposes. Similarly, “the sum of angles in a triangle is 180 degrees” is tautologically true only because euclidean geometry pre-defines that triangles have three sides and that there are 360 degrees around a point. In this sense, mathematics resembles a formal game of logic, where conclusions follow necessarily from axioms. This implies that mathematics is purely analytic instead of synthetic — true by definition within its own symbolic framework — rendering it impossible to be synthetic a priori.

Even disregarding all the human-defined symbols and terminology in mathematics and viewing it as platonic knowledge — pure concepts that exist beyond human experience and can be discovered through reason — they still do not provide new information. The laws of mathematics are not discoveries about the world but definitions that govern how mathematical concepts interact. The concepts of numbers and operations are predetermined by the rules of mathematics. When we say “7 + 5 = 12,” we are not uncovering a new relationship but simply following the logical implications of the system. Kant’s claim that mathematics is synthetic a priori rests on the assumption that mathematical truths reveal something new about the world. However, this overlooks the fact that mathematics is a closed system where all truths are derived from initial axioms and definitions. Mathematical truths are tautologies, statements that are true by virtue of their logical form. They do not provide new information because their conclusions are already implicit in their premises. Therefore, mathematics is an example of analytic truths — true by definition within the system of rules that govern them.

So far, this essay has focused closely on debunking the supposed synthetic nature of mathematics. However, even its “a priori” nature is inaccurate as mathematics is entangled with experience. While Kant argued that counting and geometric intuition are innate, these capacities likely emerge from evolutionary and cultural practices. For example, the concept of “number” arises from the human need to quantify objects, a skill honed through millennia of interaction with the physical world. Similarly, Euclidean geometry codified observations about land measurement in ancient societies. Far from being pure intuitions, mathematical systems are refined abstractions of empirical regularities. Human’s learnt arithmetic by counting apples, trading goods, and counting casualties, and ancestors generalised these operations into formal rules. As such, mathematical knowledge is reliant on experience and would thus fall under the a posteriori category.

Mathematical truths are also highly dependent on the context in which it is discussed—something which requires experience to understand. Consider a computer operating in base-8: when asked to compute “7 + 5,” it would interpret “7” and “5” within its base-8 framework, yielding “14” (equivalent to 12 in base-10). Here, “7 + 5 = 12” is false, demonstrating that arithmetic truths depend on the contingent rules of the system in use. Conversely, when people hear arithmetic statements like “7 + 5 = 12”, we take that as tautologically true without much hesitation. This is because through our education, day-to-day experiences, and the fact that we have 10 fingers, we have assumed base 10 to be the default mode of arithmetic. Similarly, non-Euclidean geometries—such as Riemannian geometry on a spherical surface—reveal that the sum of a triangle’s angles can exceed 180 degrees. Once again, since it is much more common for people to interact with geometric shapes in two-dimensions, we tend to be unaware of spatial anomalies. These variations undermine Kant’s claim that Euclidean geometry reflects a universal a priori intuition of space. Instead, they highlight how mathematical “truths” are relational—valid only within specific frameworks—and shaped by empirical discoveries. As such, mathematical statements are instead a posteriori, since it is knowledge derived from our lived experiences.

In conclusion, Kant’s characterisation of mathematics as synthetic a priori knowledge does not withstand scrutiny. While he argued that mathematical truths like “7 + 5 = 12” or “the sum of angles in a triangle is 180 degrees” are both informative and independent of experience, this essay has put forth arguments that demonstrate that mathematics as analytic and empirically grounded. The tautological nature of mathematical truths—true by definition within their own systems of rules—renders them incapable of providing new information, disqualifying them from being synthetic. Furthermore, the dependence of mathematical concepts on human-defined conventions, such as base-10 arithmetic or Euclidean axioms, reveals that these truths are not universal but contingent on the frameworks we adopt. Even the supposed a priori nature of mathematics is undermined by its reliance on experience, as mathematical systems emerge from evolutionary, cultural, and empirical practices. Ultimately, mathematics is not a realm of synthetic a priori knowledge but a dynamic, context-dependent tool shaped by human interaction with the world. By dismantling Kant’s framework, we recognise that mathematics is neither synthetic nor purely a priori but a product of both logical necessity and lived experience.

Acknowledgement

I would like to thank my instructor, Professor Kuldip Singh, for his continued guidance, patience, and insightful discussions throughout the course thus far. Without whom and which, the completion of this work would not be possible.

Works Cited

Immanuel Kant - Critique of Pure Reason, 1781