Kant’s Synthetic Mathematics In Light of Current Times
Kant develops his idea of mathematics early in the Prolegomena to any Future
Metaphysics1 and finds that it is both a priori and synthetic. He goes on to cite the source
of its synthetic nature as the pure forms of intuition, space and time. In recent times, a
priori and synthetic nature of mathematics has been called into question. Recent
advances in scientific knowledge and theory, greater cultural awareness and
philosophical trends have all attempted to remove the trait of synthetic from mathematics.
They do this by attacking the synthetic quality at its roots, the forms of intuition. These
forms of intuition are a cornerstone of Kant’s philosophy, and it is through them that the
structure of mathematical propositions as synthetic will crumble.
When viewing Kant’s work, it is essential that his terms be clearly understood. In
the discussion of Mathematics, a priori and a posteriori values are pivotal as well as
Kant’s definitions of synthetic and analytic. When an item is described as a priori, it is
necessary to our way of thinking and experience. An a priori truth is a truth that is
necessarily true. A priori claims do not need sense-experience as a reference. Such items
are engrained within our lowest levels of thought. We would not experience the world in
the way we do without these traits. It is important to note that Kant also includes items
that are learned from experience as possibly a priori. The subject matter for a judgment
All quotations from Immanuel Kant's Prolegomena to Any Future Metaphysics are
taken from the Prolegomena to Any Future Metaphysics: With Selections from the
Critique of Pure Reason, ed. Gary Hatfield, (Cambridge: Cambridge University, 2004),
pp. 5-134, and will be indicated in parenthetical references by the letters PR
(Prolegomena) followed by the specific page number quoted.
may be empirical but if no further experience is needed to make the judgment, it is still
considered a priori.
Items that are deemed a posteriori are items that require information to be learned
from an outside source, sense-experience. An a posteriori claim is one that you must
look outside the basic constructs of your mind for the answer. Most claims we
experience on a day to day basis in the world are a posteriori. “How many frozen pizzas
do I have in the refrigerator?” is a question that will have an a posteriori answer. I
cannot find the answer by looking inside thought processes; I must go to the fridge and
check through sense-experience. “Unlike the distinction between the a priori and the a
posteriori, which he inherited and modified in few or no essentials while providing a
radically new theory about the a priori, the distinction between analytic and synthetic
judgments is, historically, what he said it was.”(228, Beck) Since Kant is the originator
of these terms, there is little to dispute on how they are used. “Analytic judgments say
nothing in the predicate except what was actually thought already in the concept of the
subject, though not so clearly nor with the same consciousness.” (PR, 16) Analytic
judgments are often associated with definitions, and restating. The statement that “I am a
bachelor”, followed by the analytic judgment that “I do not have a wife” is a good
example. The fact that I do not have a wife is contained within the definition of the word
“bachelor.” It is impossible to affirm the first statement, and then reject the second
statement because such a case would be a contradiction. “All analytic judgments rest
entirely on the principle of contradiction and are by their nature a priori cognitions” (PR,
17). Analytic judgments do not require outside experience to formulate, so there are no
analytic a posteriori judgments.
Synthetic judgments are the opposite counterpart to analytic judgments. While
analytic judgments use exclusively the principle of contradiction, synthetic judgments use
other principles. Kant separates synthetic judgments into three groups: judgments of
experience, mathematical judgments and properly metaphysical judgments. Judgments
of experience are the clearest to understand, as they are the best example of an opposite
to the analytic judgment. A judgment of experience looks to sense-experience to
reinforce a claim. These judgments are naturally a posteriori because of this.
Mathematical judgments will be explained further in another section, but they are
considered by Kant to be both synthetic and a priori. Properly metaphysical judgments
make up the last group, and it is the existence of these judgments that Kant eventually
hopes to prove. Metaphysics must make synthetic judgments in Kant’s view, because
otherwise it would simply be a constant redefining of terms with no progress in
Kant argues that mathematical judgments are synthetic and a priori, and that the
principle of contradiction is not sufficient to explain their nature. Synthetic arguments
can, and in fact must agree with the principle of contradiction. Kant first claims that,
“Mathematical propositions are always a priori and not empirical judgments, because
they carry necessity with them, which cannot be taken from experience” (PR, 18).
Mathematical propositions are always necessarily true, not simply for a specific
experience based instance. Kant then analyzes the mathematical proposition 7 + 5 = 12
and attempts to show that it is neither analytic, not contradictory, and is thus synthetic.
The proposition follows with the principle of contradiction; in no part does it go against
itself. It would also seem to have problems from an analytic system because you do not
reach 12 by analyzing 7 or 5 at a deeper level. The result of 12 is not inherently known,
this can be more clearly seen when larger numbers are used. “One must go beyond these
concepts, in making use of the intuition that corresponds to one of the two [numbers]”
(PR, 18). No sum can be created from a simple analysis; further work must be done
before it can be created. The intuition must be used to work upon the numbers, creating
representative instances in the brain that can be worked with to find the answer. Kant
then states that pure mathematical cognition, “must therefore go beyond the concept to
that which is contained in the intuition corresponding to it, its propositions can and must
never arise through the analysis of concepts, i.e. analytically” (PR, 20). The
representations in the brain, concepts, are not sufficient in themselves. There is a level of
intuition lying deeper within the brain that acts upon the concepts. The resulting sum of
this action is now new information, not previously contained within the individual
numbers 7 and 5.
Kant goes into this idea of the intuitions being the source of mathematical
cognition in a later chapter. He finds that mathematical cognition “has this distinguishing
feature, that it must present its concept beforehand in intuition and indeed a priori,
consequently in an intuition that is not empirical but pure” (PR, 32). The intuition
described cannot be based in concepts or on experience and this seems to be a problem
for Kant. “But how can the intuition of an object precede the object itself?” (PR, 33)
Sense experience cannot be the cause, but the form of sensory intuition could provide an
answer. The senses create a construct of the world, not as it is, but as it appears to us.
This constructed version of the world has certain requirements for us to comprehend the
sensory data. These requirements are commonly recognized as time and space. Kant
refers to time and space as pure intuitions because they precede experience itself.
These pure intuitions are vital to Kant’s creation of mathematics as synthetic,
without them, the structure of Kant’s argument falls apart and math is revealed to be an
analytic process. Mathematics is considered synthetic by Kant because the intuitions
provide an alternative mechanism outside of the principle of contradiction. He agrees
that mathematics must necessarily agree with the principle of contradiction already, but
presses that there is more going on than just that. If it were shown that these intuitions
were false creations, there would be nothing more to mathematics than the principle of
contradiction, and it would have to be considered an analytic process.
Philosophers have been debating the validity of mathematics as synthetic from the
time Kant proposed the idea to current day. It is a commonly held belief that Hume’s
philosophy of mathematics was opposed to that of Kant. Kant himself felt this way, “he
by no means made a classification of propositions as formally and generally, or with the
nomenclature , as I have here, it was nonetheless just as if he had said: Pure mathematics
contain only analytic propositions” (PR, 20) Not everyone feels this way however, as
R.F. Atkinson states in his essay on Hume’s Mathematics, “…whilst Hume indubitably
held mathematical propositions to be a priori and necessary, his apparent conception of a
necessary proposition was wider than the current –or indeed Kant’s --- conception of an
analytic one” (Atkinson, 127). It is a difficult argument to clearly solve, because Hume
did not use the same definitions and terms that Kant did. Kant’s terms of Analytic and
Synthetic are very specific in how they are formulated, and to say that Hume though of
mathematics in either of these two terms would be misleading. Other groups of
philosophers have objected to the very notion of a priori statements across the board. The
Logical Positivists believe that a statement is true only if it is empirically verifiable.
“The denial of the synthetic and a priori character of mathematics, not only by logical
positivists but also by the majority of contemporary mathematicians, is the greatest set-
back Kantianism has suffered in the last 150 years” (Stern, 531).
When Kant’s synthetic mathematics is addressed from a wider cultural view, it
comes up short on several levels and its validity is called into question through another
means. Kant creates his ideas of the forms of intuition from the point of view of a highly
educated individual. The society he was born into had a history of mathematics and
geometry, and he was taught these concepts at a relatively early age. This upbringing
stands in contrast with many others, and limits Kant’s understanding in an unexpected
manner. Mathematics is a learned ability, something that is taught in schools and other
institutions, it is not necessarily something innate. If it was entirely innate we would not
have to teach counting and addition in early schools. Mathematics, once learned, changes
your view of the world in a way that is irreversible. Those that do not know math view
the world in terms of single, few or many. These abstract terms are sufficient for many
lifestyles. Kant’s culture affected the structure of his philosophy, so that he could no
longer think outside the structure of mathematics. In response, Kant might refer to the
fact that the algebra that is learned is based off the innate sense of time. Kant views time
as a steady, continuous river and fits arithmetic along with this view. Because time is a
sequential, never-ending process, algebra seems a perfect fit within its structure.
Unfortunately, not all cultures view time this way. Many religions and cultures view
time as a cyclical process. Arithmetic does not meld so well with this view of time.
Viewing other cultures creates problems for Kant’s view of mathematics and time.
Modern mathematics provides yet another route to take when reviewing Kant’s
synthetic mathematics. Kant wrote the Prolegomena in a time when Euclidian geometry
was all that was known, and Newtonian physics was still fairly fresh. Since that time,
there have been advances in mathematics and science as a whole that create new
problems for Kant’s forms of intuition. With General Relativity, curved space became a
reality, and time is relative. Quantum mechanics shows that on the most basic level,
space and time operate in a manner completely alien to our way of thinking. String
theory introduces the idea of many more dimensions than those we currently experience,
and it is all done through complicated mathematics. Suddenly the reality we perceive,
albeit one seen through technological experiments and mathematical theory, is different
from our common notions of space and time. Kant holds that it is impossible to get at the
things-in-themselves, and our intuitions of space and time create the appearances we
perceive. Then how can the very product of these intuitions, mathematics, create such
theories that seem to be in conflict with its very source?
It is difficult to find a complete response to this problem, since Kant was not
around at a point where it could ever have been proposed. He does however make
reference to a problem that has a correlation to the problem of modern mathematics, that
of space being either substantival or relational. This debate occurred between Leibniz
and Newton a bit before Kant’s lifetime, but the issue was surely still in contention.
Newton believed space was a substance that physical objects moved through while
Leibniz objected; saying space is only comprehensible on a basis of interrelating objects.
Kant sided with Leibniz, “They did not realize that this space in thought itself makes
possible physical space, i.e. the extension of matter; that this space is by no means a
property of things in themselves, but only a form of our power of sensory representation;
that all objects in space are mere appearances” (PR, 39). He continues on to say, “these
appearances must of necessity and with the greatest precision harmonize with the
propositions of the geometer, which he extracts not from any fabricated concept, but from
the subjective foundation of all outer appearances, namely sensibility itself” (PR, 39).
The geometer, in using geometry cannot be incorrect in his findings because he goes
beyond the concepts to sensibility. The appearances cannot be out of accordance with the
mathematics according to Kant. The problem of modern mathematics seems a daunting
task for proponents of Kantian mathematics to overcome.
At the heart of Kant’s philosophy in the Prolegomena to Any Future Metaphysics
is the idea that Mathematics is a priori and synthetic. He reasons that for Metaphysics to
have value, it must be a priori and synthetic, and Math provides another instance of these
qualities. Math is important to Kant’s philosophy in that it lays the foundations to his
explanation of human thought processes, as well as reinforcing the early belief that
Metaphysics has some merit. Without the synthetic quality of mathematics, Kant’s entire
philosophy suffers a significant challenge to overcome. The forms of intuition provide
the most fertile ground to attack the quality of synthetic mathematics, and it seems fairly
clear that upholding that quality is an uphill battle for Kantians.
List of Works Cited
Stern, Alfred. “Kant and Our Time” Philosophy and Phenomenological research 16.4
(Jun., 1956): 531-539. JSTOR The Robert E. Kennedy Library, California Polytechnic
State University. 30 November 2005 <http://www.jstor.org>
Atkinson, R. F. “Hume on Mathematics” The Philosophical Quarterly 10.39 (Apr.,
1960): 127-137. JSTOR The Robert E. Kennedy Library, California Polytechnic State
University. 30 November 2005 <http://www.jstor.org>
Beck, Lewis White. “On the Meta-Semantics of the Problem of the Synthetic A Priori”
Mind: New Series 66.262 (Apr., 1957): 228-232. JSTOR The Robert E. Kennedy
Library, California Polytechnic State University. 30 November 2005