Kant sought to address a flaw in Hume's view that knowledge could only come from empirical matters of fact or rational connections of ideas known a priori. Kant introduced a new class of propositions called "synthetic a priori" propositions. These propositions expand knowledge through connections of ideas that are known independently of experience. Kant argued pure mathematics contains synthetic a priori propositions because mathematical concepts are constructed from pure intuitions of space and time that are imposed on objects a priori. This allowed metaphysical concepts to also be known through synthetic a priori propositions, resolving issues metaphysics faced.

1. Closing the gap between rationalism and empiricism
By way of the Synthetic a priori Proposition.
Robert Morien
990-41-5960
Philosophy 498
The University of Wisconsin-Milwaukee

2. I. Introduction
Detecting a flaw in Hume’s assessment of knowledge, Kant sets out to prove
that matters of fact can be known from the connections of ideas. In addition,
he goes one step further and asserts that, besides propositions being either
analytic or synthetic, there is a very special class of propositions known as
synthetic a priori propositions.
The purpose of this paper is to examine Kant’s motivations for thinking that
there exist this special class of a priori propositions which are neither
analytic nor known from logical principles alone, as well as examples of this
special type of proposition; namely, what the requirements are for a
proposition to be a synthetic a priori proposition.
II. Motivation
By definition, metaphysics must contain propositions a priori, which are
separate from any observation or sensual experience, i.e. a posteriori, since
the focus of this science contain topics which are empirically unverifiable.
Topics such as; the existence of God, the immortality of the soul, the
connection between the mind and the body, etc, all consist of non-physical
entities that cannot be demonstrated by empirical measurement. Therefore,
metaphysical propositions must come from the relation of ideas or
perceptions known as “pure” concepts and from these are propositions of the
external world immediately known.
David Hume asserted that all knowledge either consisted of these
connections of ideas (rationalism), or matters of fact (empiricism) and these
connections of ideas could be known a priori, but matters of fact must be
known a posteriori and could never be known a priori. In addition, Hume
did not hold that there was a union between these matters of fact and
connection of ideas.
His example of why we cannot know experiences about the external world a
priori was that of cause and effect; he could not see how given an x, there
must necessarily be something else y, or how the idea of such a combination
could occur a priori. From this he concluded that reason falls prey to
confusing subjective necessity; i.e. the habit of after experiencing many
events m followed by n ones, one automatically expects n to be the case,

3. with objective necessity, i.e. the necessary connection between the rational
and existence.
Hume’s argument that knowledge of matters of fact cannot be known a
priori thus goes as:
 The union of cause and effect is a necessary connection that involves
relations of ideas and matter of fact propositions.
 Only propositions concerning relations of ideas can be necessarily
true.
 Therefore, we cannot have a priori knowledge of matters of fact such
as cause and effect.
In essence, what the empiricist was saying was that a science such as
metaphysics can not exist.
Traditionally, Hume’s belief that ideas are either known independently of
experience, or are known from experience, is what has come to be known
“Hume’s fork”. This philosophical utensil was what awoke Kant from his
“dogmatic slumber” which prompted him to resolve the problem
metaphysics faced by claiming that not only can relations of ideas be known
a priori; matter of fact propositions can also be known without any
experience in the matter, and that there was a connection between these two
types of knowledge. Kant put together this thesis and formed a new branch
of philosophy, known as transcendental philosophy in the Prolegomena.
III. Analytic vs. synthetic propositions
In the Prolegomena, Kant derives the conditions for what makes analytic
and synthetic a priori propositions possible
For Kant, synthetic is the opposite of analytic, in the sense that analytic
propositions contain subject matter whose predicates are contained in the
subject, and hence do not provide any further knowledge. The propositions,
“all bachelors are unmarried men”, or “all two legged animals are biped” are
not true because of the way the world is constructed; rather they are true
because of their propositional content. As a result, analytic a priori
propositions do not provide us with any further information since in the
proposition “all bachelors are unmarried men” the predicate term is already
included in the definition.

4. A further distinction of analicity is through the law of contradiction. Since
the predicate of an analytic proposition is already included in the definition,
it cannot be denied without contradiction. To deny the proposition “all two
legged animals are biped”, is to say that “anything that is g and biped is not
biped” which is clearly a contradiction. The law of contradiction is what
preserves the truth of the analytic proposition.
On the other hand, the predicates in synthetic judgments do not contain
subject matter that is already contained in the subject. These propositions
increase our knowledge and add content to our concepts. The proposition,
“all bachelors are happily unmarried” evaluates the subject matter of the
predicate and supplies information not already known.
Up to this point Kant is in agreement with Hume regarding the distinction
between analytic and synthetic propositions, but Hume holds that synthetic
propositions can only be known a posteriori; that is we can add to our
knowledge only through empirical observation, or through experimental
inference, and cannot be known a priori. This is where Kant starts to detach
from Humean knowledge and proposes to establish the synthetic a priori
proposition.
Kant saw this incongruity between matters of fact and the connection of
ideas as a flaw in Hume’s reasoning and knew that in order to save the
science of metaphysics, he needed to resolve this inability the synthetic
propositions contained that did not allow for the expansion of knowledge
from the connection of ideas. His answer was to introduce a new set of
propositions that would allow for the expansion of a priori propositions.
These propositions he called synthetic a priori propositions.
IV. Closing the gap between rationalism and empiricism
Unlike Hume, Kant thought that not only do synthetic a priori propositions
occur; they also provide the stepping stone for much of human knowledge.
Pure mathematics and arithmetic are an example of such propositions. In his
view, mathematical knowledge consists of pure, synthetic propositions.
Pure in the sense of not proceeded from a connection of ideas, but from the
construction of the connection of these ideas; i.e. derived from a pure
intuition.

5. Given that all mathematical knowledge must arrive from the pure intuition,
it follows that the mind can create all of its concepts, and thus develops a
relationship between these a priori propositions with other mathematical
truths.
How the mind does this is through the construction of an intuition of a space
and a time, such that geometry is based upon the intuition we call space, and
arithmetic is based upon the intuition we call time. These intuitions are a
priori intuitions since they do not come from any of the senses but are
constructed only in the mind. For example, if you take everything you know
to be the case about bodies and their movements and strip away everything
that can be observed empirically, all that remains is space and time. These,
therefore, constitute pure a priori intuitions.
It is the act of counting fingers in arithmetic for example, that we
successively add units in time, whereas geometry is the act of drawing a
straight line in space.
This spatio-temporal relationship is the precondition that we impress on
objects in the sensible world and allows us to derive certain truths about
these sensible objects a priori; such as no more than three perpendicular
lines that intersect each other form the dimensions of space or a straight line
can be drawn to infinity. This pure form of sensible intuition Kant called the
transcendental aesthetic. It is this transcendental aesthetic that provides the
missing link between knowing truths about the external world and knowing
things through synthetic a priori propositions.
Empirical truths contain content that we may never know of things in
themselves; i.e. the substance of an object can never project its subject
matter into our minds; the content of that information is dependent on the
perceiver. For example, the stars and the moons one sees and perceives may
be separate and discrete from the stars and moons another sees and
perceives.
Geometrical propositions, on the other hand are not illuminations of things
in themselves, but are known by us a priori. Space is not “out there” but in
us, and the act of imposing our intuition of the concepts space and time
allows us to ascertain external truths that are necessary and absolutely
certain. For this reason we are all able to arrive at the same geometrical
truths.

6. A Kantian transcendental argument that supports these claims would appear
as something of the form:
 The sum of all angles in a triangle forms a straight line.
 That this is true can not be made known from concepts, but from the
intuition.
 Space and time are two constructs that also do not arise from
concepts, but are made known through the intuition a priori.
 Therefore, geometry, and thus mathematics contain truths that are
known a priori, and that the propositions that form this knowledge are
synthetic a priori propositions.
 That this is the case owes to the fact that in addition, these
propositions are necessary and absolutely certain.
Metaphysical propositions also must contain subject matter that is necessary
and absolutely certain, so that things may be made known in themselves.
Knowing that we can have knowledge about mathematical truths through
synthetic a priori propositions, it follows that we can have knowledge about
metaphysical truths through the employment of synthetic a priori
propositions.
For this reason, Kant had a firm conviction that his transcendental
philosophy was the complete solution to the problem regarding the
possibility of synthetic a priori propositions; and further to the problem
metaphysics faced.
The transcendental philosophy thus squeezes between the propositions of
empiricism and rationalism driving a wedge in Hume’s fork.
V. Concluding remarks
Prior to the Prolegomena, the only type of geometry known was Euclidean
geometry. Since that time other geometries have been shown to exist, such
as Riemannian geometry, where any number of dimensions in space can be
shown to exist. This poses a threat to Kant’s transcendental idealism since
his thesis implies that the intuition conditions our understanding into being
able to see and perceive external objects. In the case of Riemannian
geometry this implies that we can be conditioned to visualize spaces that
contain an infinite amount of dimensions. This obviously cannot be so,

7. since our minds are finite and not constructed in a way that enables us to
perceive an infinite number of dimensions.
One wonders what Kant’s response would have been to such a question.
Supposing that the thesis on how the intuition constructs space and time, one
could see a possibility for Kant to avoid this perplexing problem. Taking his
stand that you get out only what you put it in, it seems reasonably fair to say
that a Kantian would have constructed an argument of the form:
 Space and time are two constructs that also do not arise from
concepts, but are made known through the intuition a priori.
 The intuition is constructed by a priori independent of experience,
and within one’s own mind.
 Therefore, the propositions that we can know a priori are those that
contain only the intuitions we have put in them.

8. Works Cited
Kant, Immanuel: “Prolegonema”
http://www.earlymoderntexts.com/f_kant.html;
Kant, Immanuel: “Prolegonema”, translation by Jonathan Bennet;
Kant, Immanuel: “Critique of ure Reason”
http://www.trinity.edu/cbrown/modern/kant.html: “Kant: An Overview”
Blackburn, Simon” “The Oxford Dictionary of Philosophy”
Class notes from lecture in Philosophy 498