In The Meno Plato presents an argument for innate mathematical knowledge. Socrates draws a square with a diamond inside on the ground in front of a slave boy who has had no mathematical teaching. Socrates then draws a diamond inside the square. The slave boy is able to explain, and correct himself when making a mistake, that the sides of the 8ft square are equal to the length of the diagonal of a square that is 4 square feet.
Since the slave boy had no mathematical education, the conclusion Plato draws is that he must have been born with the knowledge. He then argues that the only way to make sense of how that knowledge got there is if we had an existence prior to birth, a soul which existed in a world of forms. There, it apprehended perfect mathematical forms, gaining mathematical knowledge, before becoming trapped in this world of appearances.
An empiricist could respond that we gain the concepts of number and shape from experience and then gain mathematical knowledge when analysing those concepts. The slave boy may not have had any formal mathematical education, but he must have experienced the shapes of objects in his life. That experience could be how gained geometric concepts which he then used to demonstrate mathematical knowledge. Socrates’ questions could also have helped to sharpen and develop those concepts.
Integration: The knowledge demonstrated could therefore have been gained by analytic a priori reasoning about concepts gained from experience. That being the case, the knowledge demonstrated by the slave boy cannot be used to prove that there is innate knowledge. We have a better explanation, that it was caused by experience and therefore have no reason to accept the conclusion that there is innate knowledge.
Locke vs Leibniz on innatism
Locke’s critique of innatism
Locke believed there is no innate knowledge as the mind is was ‘tabula rasa’ (blank slate) at birth. In Locke’s time there was debate about whether propositions such as ‘something cannot be and not be at the same time’ or ‘whatever is, is’ were innate.
Locke’s argument has two parts:
1: If a proposition were innate then it would be universally assented to, meaning everyone would be know it because everyone would be born with it. Universal assent is therefore a necessary condition of innate knowledge. However, children and idiots do not know the supposedly innate propositions.
Integration: So, there is no knowledge which satisfies the necessary condition for innateness of universal assent and thus there is no innate knowledge.
2: Even if there was a proposition that was universally assented to, that wouldn’t necessarily make it innate. There could be some other explanation of how everyone came to know something other than it being innate. For example, there could be something universal about experience in general, or about the experiences which people happen to have had, which produced the knowledge.
Integration: So, universal assent is a necessary but not a sufficient criteria for innate knowledge. In order to show that a proposition is innate, it is required to show that it could not have come from experience.
Leibniz’ response to Locke
Leibniz responded to Locke’s ‘children and idiots’ augment by claiming that people can have knowledge even if they can’t express it. We can tell by the way children and idiots behave in the world that they must ‘unconsciously’ assent to such necessary truths as ‘something cannot be and not be at the same time’.
Furthermore, Leibniz argued that innate knowledge could require some sort of unlocking or triggering process by interaction with experience in order to be consciously known. Leibniz illustrated this with a block of veined marble where the veins happened to be along the outline of a statue of Hercules. A stonemason would merely need to hammer the block a little and the marble would break away along the veins, revealing a statue. In this analogy, the hammer blow is like experience and the statue is the innate knowledge.
No one would imagine that a statue of Hercules could be created by one hammer blow merely from one hammer blow. The block of marble must have had an innate potential in order for that result to occur.
Similarly, it could be that when experience hits our mind, it unlocks an innate potential for knowledge. It might look like experience is causing the knowledge but really it is just activating the pre-existing innate knowledge. Importantly, on this view, although experience plays a role in our knowing of these propositions, the relationship between experience and knowledge is not, as empiricists claim, one of inference from experience to knowledge.
The key aspect of this argument is that we know that a single hammer blow cannot by itself create a statue from ordinary blank marble.
However, the question then is what could there be about experience and propositions which we supposedly have an innate potential to know which, likewise, could tempt us to doubt a simple causal relation between them as the empiricist wants to suggest? Leibniz answers this question with the following:
Leibniz’ argument from necessary truths
Leibniz thinks he can prove this knowledge could not have come from experience and must therefore have been innate by pointing out that the supposedly innate propositions are necessary, as they cannot fail to be true. However, we only ever experience contingent things. All the objects in our experience depend on something else for their existence.
We could not have inferred or gained knowledge of a necessary proposition from merely contingent experience.
This is because experience of contingent beings does not involve necessity, so cannot be used to infer necessary propositions. No amount of contingent experiences, such as only ever observing things either being or not being, can provide us with knowledge of a necessary proposition, such as that something cannot both be and not be at the same time. Just because we have always experienced something either being or not being, that doesn’t mean that it must be the case that a thing either is or is not. Necessary propositions are those which must be true. They cannot be gained from experience.
If knowledge of necessary propositions cannot be gained from experience then they must be gained through the unlocking of our innate potential to know them by experience.
The mind as a ‘tabula rasa’
Locke argued that the mind is ‘tabula rasa’ at birth – a blank slate with no innate concepts. He argues all our ideas come from either sensation or reflection. Sensation is when our senses experience objects in the external world. Reflection occurs when we experience the ‘internal operations of our minds’.
However, the sensation of yellow is quite different from the concept of yellow we gain from reflection. Locke’s distinction doesn’t account for that. Hume proposed a different set of distinctions to solve this. Everything our mind directly experiences is a perception. Perceptions from the external world are impressions, perceptions from inside our mind are ideas. Ideas are usually less forceful and vivid. Something remembered is less vivid than the original experience (except in special cases like madness). This is because ideas are ‘faint copies’ of impressions.
Ideas can be simple or complex. A simple idea is one which cannot be broken down any further, e.g redness. Complex ideas are combinations of simple ideas. A red rose is a complex idea made from the combination of the simple ideas of redness, the smell of a rose, the texture, and so on.
Hume’s copy principle argues that all simple ideas are copies of impressions. All ideas in the mind therefore ultimately derive from impressions. Therefore, there are no innate concepts. Hume’s proof is to challenge anyone to think of a simple idea that was not copied from an impression. He also points out that lacking an impression results in a lack of the concept, e.g. someone blind from birth lacks the concept of color.
The missing shade of blue. Hume himself puts forward an exception to the copy principle, however. Someone who had never before seen a certain shade of blue would be able to conceive of it were they showed a list of all the other shades of blue, with the one they had not seen missing. The idea of that shade of blue would then not have been copied from an impression. Hume himself suggests this example is not significant, but arguably it is.
To defend Hume, one could claim ‘shades’ of blue are really complex ideas – mixtures of the simple ideas of blue, black and white. In that case, the missing shade of blue is conceived by the imagination in the same way as a golden mountain is conceived by joining the simple idea of gold and complex idea of mountain.
Philosophers disagree about how to break down complex ideas like knowledge, justice & beauty into simple ideas however. Either that is because the ideas and the method of acquisition of them are so complex or it’s that Hume’s theory is wrong.