John Stuart Mill:

A System of Logic, Ratiocinative and Inductive

Is it necessarily true that 2+1=3?

Background

John Stuart Mill (20 May 1806 – 7 May 1873) was the son of James Mill who was good friends with Jeremy Bentham. Jeremy Bentham helped to educate John Stuart Mill and Mill was given an extremely rigorous education. He went on to become one of the most influential thinkers and founders of Liberalism.

Mill wrote his work A System of Logic, Ratiocinative and Inductive before his other works on ethics and politics for which he is perhaps more famous.

He wanted to provide an interpretation of logic that was entirely empirical, that is based on evidence of the senses. If we haven’t experienced something, he thought, how could we know it? He was not very persuaded by the explanations that we popular at the time: for example, the idea that we might understand these things because we were born with them stamped on our minds (Descartes) or because the world had to conform to our mind (Kant). Mill thought that the fact that there could be no external evidence proving such ideas meant that anything could be justified by these ideas, which he called intuitionism.

Verbal and Real propositions

A statement is a Verbal statement if it only tells us what words mean, but doesn’t give us any information about the world. For example, if I say that ‘a proposition is a statement saying that something is true or false’ then Mill thinks I’m not telling you anything about the world, I’m just telling you what the word proposition means. Mill thinks that such statements don’t have any content, that they are empty. They are just the rules for the use of symbols. E.g. if I say, A=B I haven’t actually told you anything about the world, I’ve just given you a rule to follow.

Real statements, on the other hand, tell us something about the world and how things are related to each other.

A good metaphor for this distinction is a map. The key of a map are Verbal propositions. It doesn’t really matter what symbols we use for ‘Mountain Peak’ for example, so long as we follow the key. The map itself are Real propositions. They tell us about the world. If I say, ‘the mug is on the table’, I am providing you with a (albeit boring) ‘map’ of the world.

Associationism

JS Mill was an Associationist. He thought that we learnt about the world by experiencing two things together repeatedly, and therefore we connect or associate them with one another. For example, if every time I let go of a pen, it falls, then I will associate letting go of a pen with pen falling. This was, he thought, how we come up with our real propositions.

What about mathematical and logical propositions?

The difficulty for Mill was with Mathematical propositions. Are mathematical statements, like 2+1=3 verbal or real? Mill actually thought that they were real. He thought that they tell us something about the world. Therefore, Mill thought that we learn things like 2+1=3 through experience and association: we just find that in our experience 2+1 tends to equal 3.

Similarly, Mill thought that an argument like the following is not necessarily true either:

All men are mortal
Socrates is a man
∴ Socrates is mortal

He thought that the first, categorical, premise is simply an induction: every man we’ve ever met is mortal, and therefore we conclude that all men are mortal.

—but this suggests that 2+1=3 (and other mathematical or logical truths) aren’t necessarily true - that one day we might come across a situation where 2+1=4! That seems ridiculous to most people— But not to Mill! Mill thinks that the necessity of maths and logic is just an illusion!

Discussion

Can it really be the case that we might one day find an example where 1+1=3? If someone came to that conclusion, would you want to see the example or would you just conclude that they were wrong?

Interestingly, many historians of maths think that mathematical rules often began as being only probably true: It seems to be the case that the Ancient Sumerians and Egyptians were aware of the Pythagoras’ theorem (before Pythagoras was born) but for them it only happened to be true. If for example, an ancient Sumerian fence-maker was measuring a triangular field and it came out as a²+b²=c²+1, they wouldn’t have been surprised or have believed that they’d made a mistake in measurements. a²+b²=c² was just a rule of thumb, something that generally worked. It was only with the Ancient Greeks – Thales, Pythagoras etc. that people began to provide mathematical proofs – reasons that showed why a²+b²=c² and such like. Thus, after Pythagoras, if a fence-maker found that a²+b²=c²+1, an assumption would be made that one of the measurements must have been wrong.

According to this version of history, these regularities that only happened to be true are now held firmly in place by a new set of activities – acts of proof, logical connections, and categorisations. What has changed here is the nature of the warrant and the conclusion.

From experience, the conclusion was only probably true, but with the proof it became necessarily true.

A nice way to visualise this is as three sides of a triangle. In the first instance, the sides were loosely held together by experience, but in the second case they were fixed by the proof:

Tasks

Explain Mill's point of view:
- Explain Mill’s distinction between verbal and real propositions in your own words.
- Explain why Mill believed that mathematical and logical propositions were not necessarily true (use the word association in your answer).
Give one objection to Mill's argument.
Do you agree with Mill? Do you think it is, at least in principle, possible for 2+1=4?
Extension: Think of an example for each of Mill’s methods

Extension: Mill’s Methods

Mill also came up with five methods for testing our associations and establishing strong connections between causes and effects. They are a bit like argument forms. Below each quotation is the method written out algebraically, where each letter stands for some feature or variable. So, for example, in the the first method the Direct method of agreement, imagine that A stands for ‘Oxygen was present’ and w stands for ‘a fire was started’ (and B, C, D, E, F, G, all stand for other chemicals being present) you’d be justified, according to Mill in concluding that the presence of oxygen is necessary for there to be fire.

1. Direct method of agreement

If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.