The Problem of Induction


Can We Acquire Knowledge?

Clemens Lode, Apr 2015*

In his Critique of Pure Reason1, Kant wished to argue against the empiricism of David Hume. He claimed that induction in relation to causality could not be a means of learning anything about nature since the justification of the validity of induction would in turn require induction. According to Hume, to posit that the identity of an entity at a future point in time (without external influence and with attention paid to internal processes) is the same as in the present is not valid.

Kant attempted to solve this problem by creating the term synthetic a priori statement2 By that, he intended to show that there are statements about the world which would not require induction:

Analytic statement: A statement whose assertion is given by the concept of the subject. As a result, measurements are not necessary to determine whether it is true or not (e.g., “Triangles have three vertices”).

Synthetic statement: A statement whose assertion is given not alone by the concept of the subject alone; i.e., measurements are required to determine whether it is true or not (e.g., “This form has three corners”).

A priori statement: A statement which can be substantiated independently of experience (e.g., mathematical statements).3

A posteriori statement: A statement which must be substantiated through experience (e.g., “Bodies are heavy”; we must first lift a body to determine its weight).5

In his work, he sought synthetic statements which were at the same time a priori statements and, as a result, could be substantiated without empirical knowledge of reality. His very lengthy explanation in Critique of Pure Reason did not help to clarify what he—knowingly or unknowingly—actually meant by his notion of analytic and synthetic statements, as well as by the distinction of a priori and a posteriori statements.4

The point is that his synthetic statements concern nothing other than measurements A synthetic statement is thus nothing other than a statement about the effect of a representation of a concept—an entity. For instance, the statement “All chairs are made of material” refers to a property of the concept “chair,” while the statement “All chairs are made of the material wood” relates to the tangible effect of the property “material.” A synthetic a priori statement thus would be nothing other than a “statement whose assertion is not given by the concept of the subject (i.e., a measurement!), but can be substantiated independently of experience (i.e., not a measurement!).” A measurement that is not a measurement is obviously a contradiction; for this reason, by the Axiom of Identity, synthetic a priori statements cannot exist.

By the Axiom of Identity, Kant’s synthetic a priori statements cannot exist. That means that there are no statements that can be shown to be true without induction.


Hume concerns himself with the future, and hence with the question of whether knowledge we acquire about the world can be applied to future events. “Time,” however, is ultimately merely a construct of the mind.7 In more general terms, it deals with the question of whether knowledge acquired from a past situation is also valid in a situation at different point in time. Still more generally, we can bring under criticism the use of concepts universally:

If we have established, e.g., that when dropping an apple, it falls downward, who is to say that the same must also hold for a different apple (or at a different but comparable location or point in time)?

Possible answers to this problem could be that we may have erred in constructing the concepts in question, and there are still many more significant properties we might have not yet discovered. Or, there could be a coincidental external influence, e.g., a strong gust of wind could blow the apple upward.

But that is not what Hume aims at; he is concerned about the validity of concepts, i.e., whether we can acquire general knowledge about the world when we exclude such special cases. We have defined the term “concept” in this way for the very reason that it includes entities that, for example, possess the property of falling downward. It makes no difference, whether we now consider other apples in our fruit basket or apples existing far in the future. In both cases, we speak of the same concept, “apple.” If future apples possess other properties than our present apples, we must diversify our concept “apple.” When defining the concept, we have to either restrict the selection of entities or include a dynamic component which adds, e.g., the factor of time into the description of the properties. Exactly such a discussion is currently going on in the sciences concerning the gravitational “constant.” If, for instance, in the future, the gravitational constant should change, it would say nothing about the validity of concepts per se but instead would speak to our potentially incomplete concept of gravitation where we should have included a change of the gravitational constant depending on the time and location.

Hume’s problem of induction is ultimately aimed at the fact that we are not omniscient when we establish concepts.

Ultimately, we see that Hume’s argument is a matter of nonexistent omniscience in the establishment of concepts. We can, therefore, compare him with Kant’s “thing-in-itself”: potentially, there is always a level further on, a (still?) unknown “true reality,” which was as yet unknown to us when we defined our concepts.

We could also formulate the question in more general terms: does carrying out a deduction depend on empirical facts, i.e., can we perform experiments which can determine whether we can determine things? This approach leads to an endless cycle of questioning (a so-called recursion, see below)—to answer the question we must be able to answer the question. It has come to this recursion, since we cannot ask any questions which bring into question the presupposition for the question—this would be the fallacy of the stolen concept. We could not then bring into question the validity of concepts if we pose a question that uses concepts. At most, what we owe to Hume is that we should not assume that we are omniscient; we should require proof for scientific theories and re-examine existing knowledge when gaining new insights.


Self-reference (recursion): If a statement or a process references itself, it is called recursive.

Working step-by-step through, e.g., a recipe usually is not recursive. But if there is a task like “Add some flour to the dough. Knead the dough. If the dough is sticky, we are complete. Otherwise, we add some more flour and knead it again…”, then we have a (possibly infinite) telescoping of the same process. Or imagine a photo of a person who holds up that very photo to the camera. A third example of a recursion would be cell division processes within a life-form. This is especially visible when looking at, e.g., tree branches or our system of blood vessels.

In that regard, you cannot say, e.g., “I cannot make objective statements” because that is an objective statement. And trying to rectify it by saying “I cannot make objective statements except for the statement ‘I cannot make objective statements’ ” is recursive because you would need infinite time to make an objective statement about objective statements. It is like saying that “something is true because because because because because …” without really providing a final argument. This is one of the many issues of the application of language which we will discuss in the upcoming first book of the series “Philosophy for Heroes.”


[A comprehensive discussion of this position and its political consequences for subsequent philosophers can be found in The Ominous Parallels: The End of Freedom in America by Leonard Peikoff (978-0452011175). Especially recommended is another discussion about rationalism, empiricism, and objectivism spanning several chapters in Understanding Objectivism: A Guide to Learning Ayn Rand’s Philosophy, p. 209 – 307 (978-0451236296).]
2. [vgl. S. 55 – 67 Kritik der reinen Vernunft, 978-3-86647-408-6]
3. [Closely connected with this topic is the question about a priori knowledge, which we will discuss more closely from a scientific position in the second book of the series “Philosophy for Heroes”. Philosophically viewed, the issue is clear: we have ultimately defined knowledge such that new knowledge can be formed only from existing knowledge or from perceptions of reality. Without ever having made a perception we can thus never acquire knowledge.]
4. [On the other hand, analytic a posteriori statements do not exist.
5. [Analytische a posteriori Aussagen gibt es dagegen nicht.]
6. [vgl. S. 62ff Kritik der reinen Vernunft, 978-3-86647-408-6]


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