Re: Rationalists
We all agree that a straight line is the shortest distance between two points.
How does the rationalist explain how we know this? For them it's an a priori principle but Kant shows that it's actually not one principle but the synthesis of two concepts (a) points (qualitative--from sense) and (b) distance (from where? We don't see 'distance').
Think of the analogy with 7+5=12. When you see '7' '+' and '5' the concept '12' isn't contained in those other 3 concepts they way 'unmarried' is contained in the concept bachelor. To arrive at '12' you had to do a mental operation.
The problem for rationalists is explaining where new knowledge comes from (i.e., that isn't analytic). (a) Deduction can't produce new knowledge. Also, (b) they are mistaken in believing that all a priori principles are purely analytic.
If I say to you "a straight line is the shortest distance between two points" "shortest distance between two points" isn't contained in the concept "a straight line". To arrive at the answer "the shortest distance between two points" you have to perform a mental operation to arrive at a new concept that wasn't contained in 'straight line'. You have new knowledge! Notice also that 'points' are qualitative concepts and distance is a quantitative concept. Just as we do in 7+5=12, we count on our mental digits to discover (the new knowledge) that the shortest distance is a straight line.
But, if all a priori knowledge is analytic, how do we explain how we learned something that wasn't analytic or empirical? There must be another category of knowledge: synthetic a priori. But hold on. Where does that knowledge come from? Distance is a geometric concept and so has to do with space. But, on the rationalist view we can't have knowledge of stuff that comes in through the senses so it couldn't have come from there. So where did it come from? It comes from the structure of our mind. Our mind adds spacial concepts to experience. It's how we structure experience. The new knowledge is the result of synthesizing concepts one taken from experience (points) and one taken from our mind (distance/space).
Re: Empiricists
The main problem for Locke is that on his view geometric truths (axioms)--truth about the nature of space will only be contingently true. But we think that geometric axioms are necessarily true--they are true for all time and in all possible circumstances--they can't possibly be false (think Descartes).
"Experience teaches us that a thing is so and so, but not that it cannot be otherwise."
Consider: For Locke how would we know that the shortest distance between two points is a straight line? We would learn this through experience. We'd observe instances of two points then observe that in each instance the shortest distance between them was a straight line. We'd then abstract from all the particular instances to the general claim that "the shortest distance between two points is a straight line."
Why is this a problem? Because we want "the shortest distance between two points is a straight line" to be a necessary rather than contingent truth. What the heck does that mean?
Let's take a step back. Suppose I've seen 100 ravens in my life and they were all black. I might make the inference that all ravens are black. Hold on, you say. It's possible (remember Descartes) that there might be a raven that isn't black. Fine, I say. I'm going to look at 1 million ravens then I'll conclude that all ravens are black. Can I still make this claim? Even after viewing 1 million ravens it's still possible that there exists a raven that isn't black.
No amount of raven viewings can make it necessarily true that all ravens are black. It will always be possible for there it exist a raven that isn't black. The technical word is contingent: That all ravens are black is a contingent fact (it just happens to be that way) rather than necessary fact because it's possible that there exist ravens that aren't black. Recall from Descartes that sensory knowledge is unreliable. Empirical beliefs can turn out to be false because either (a) we misperceive something (i.e., our senses deceive us) or (b) we haven't observed all instances of a phenomena.
Getting back to Locke and empiricism. If I claim to know that the shortest distance between two points as a result of observation, that fact is contingent. My knowing it depends purely on my having observed only a subset of all possible instances. It just happens to be that I've only observed the cases that conform with the claim. If observation of particular instances is how I come to know that the shortest distance between two points is a straight line then it is a contingent not necessary fact.
But mathematical and geometric (and logical) truths are necessary truths (or at least Kant thinks so). There is no possible way for them to be false. Thus, we can't know that they are true because of sensory experience. We need some other way of knowing that guarantees against any possibility of them turning out to be false. On the other hand, it can't be analytic a priori either, as we saw above. The answer is that our minds impose space and time on experience. Space and time come from us not the world.
This solves two problems. Since space and time come from the structure of our minds geometric truths will be necessarily rather than contingently true. It also solves the problem of explaining how we can have new a priori knowledge (i.e., synthetic).
Philosophical Significance
For Locke, experience writes on the mind. This is a passive picture of knowledge. Experience pours in on my consciousness. Once they’re imbedded in me I’m not an active participant. I’m only active when I later go to analyze and manipulate them. Also, our concept of space is also acquired by experience.
For Kant our experience are the product of a mixing of contributions from the mind and the real world. The real world generates the sensibilia. But there are two inputs into the mind when I experience the world. My mind imposes space and time on my experiences and generate the phenomena I perceive. On Kant’s view, you can separate these components: experiential and knower-contributing components. When you look at the latter in isolation, you’re investigating the synthetic a priori claims. Things in themselves (unexperienced by us) aren’t spacial or situated in time. This means the world as I experience it is an amalgam of these two influences (there are scholarly disputes regarding this point). For Locke, the real world is an independent reality and my subjective experience another, hence the problem of the veil of perception.
Historical significance: Prior to Kant knowers were passive. Now the knower is also at the center. We are adding space and time to the world. The real world (the world of our experience) is partly a function of what we bring into it.
Recap: Why did Kant think this is right? You start from things that can’t possibly be false (geometry) and you end up with theorems that can’t possibly false. Geometry describes space, it’s composed of necessary truths, why does this create a problem for Locke? (Space come from us, not the world). Locke can’t claim that geometry has necessary truths and it’s about space. How can you have truth about space from only perception of the world? Geometry is necessary truths. What do we know about things from experience? The experience is unreliable—this means that experiential claims are contingent. Kant’s saying there’s a contradiction here, on Locke’s view, geometry facts are contingent. But if geometry has necessary truths, it’s impossible that the characterization of space is contingent. How can we guarantee that geometric axioms and theorums be necessarily rather than contingently true? The reason we know that space truths are going to come out true is that it’s us that’s bringing the space to the table. It has to work out on Kant’s view because we’re the beings that bring the spacial framework to the table.
Definitions of Empirical vs A Priori
A priori: Only things we know completely independent of experience.
Necessary truths must be a priori because "experience teaches us that a thing is so and so, but not that it cannot be otherwise."
Strict vs empirical universality: Empirical observation can only give us assumed universality through induction. Strict universality indicates a special source of knowledge.
Certain rules like "every effect has a cause" must be necessary thus a priori.
Analytic vs Synthetic A Priori
Important: What makes an a priori proposition knowledge is NOT how it got into your head but the relationship between the subject and the predicate. A priori propositions I can know to be universally true without going out into the world to check. I know they are true by reflecting on the contents of the concepts.
Analytic: predicate is contained in the subject:
E.g.,
The black dog is black;
All bachelors are unmarried men;
All bodies are extended.
With an analytic truth, if you understand the subject (e.g., "bachelors") you can discover within it the predicate.
Synthetic: Predicate is new information but the relationship between subject and predicate is still a priori (rather than a posteriori): 7+5=12.
Locke is right that I first need to get the concepts of numbers and addition in my head by empirical means. However, I learn that 7+5=12 is necessarily true not by verifying instances of the equation in the world but by performing a mental operation. (See section below on synthetic a priori for a more detailed explanation).
The Two World Problem and Kantian Framework: All Empirical Beliefs are Synthetic A Posteriori--Not Purely Empirical
Example: Ames room, blue/gold dress
Locke says we gain knowledge of objects in the world through the senses. I come to know the world as a passive recipient of what comes in through my senses. However, there is the problem of "the veil of perception." The veil of perception is the idea that we can't step outside of our our minds to verify whether the way things subjectively appear to us are indeed the way they are in the real world. So, empiricism only tells us how things appear to us, not how they actually are.
The big Kantian move regards space and time. We experience all objects in space and time however we never 'see' space or time. Notice that it's impossible to experience an object outside of space and time. This is a problem for pure empiricists like Locke since he says that all knowledge of external objects comes from sensation. But, how then do we know about space and time?
Consider an apple. I experience its redness, its shape and so on through my senses. But I also experience it in a particular time and location in space. How do I explain how I know of its location in time and space if I don't experience those dimensions through my senses? Kant's big move is to say that our mind adds them to the experience. Space and time aren't features of the world or objects, they aren't properties, they aren't relations, they are features that our minds add to all our experiences of the world.
Test: Try to look at something in the world and perceive it outside of time and space. You can't because space and time are added to experience by your mind.
Doing Math and Synthetic A Priori
Locke is right that we learn number concepts and algebraic concepts empirically. That's how the concepts get 'into my head'. However, it doesn't follow that mathematical knowledge is empirical.
Consider: 7+5=12
I have the concept '7' and the concept of '5'. These concepts alone don't lead me to conclude '12'. I need to understand what '+' means as well. When I combine the 3 concepts my mind doesn't immediately jump to 12 (maybe now it does because you've memorized it, but when you're first learning the relations between number concepts and arithmetic operators it doesn't). I count on my 'mental digits' and arrive at 12. I perform an operation on the concepts which leads me to the concept of '12'.
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