# What is the difference between mathematical knowledge and more “general” knowledge

We know that these can all be considered as knowledges:

• 1 + 1 = 2
• free delivery for purchases over \$100
• there are approximately 100 people in this village
• Alice is dating Bob
• universe is probably started by a big bang

Mathematical knowledge tends to be very "exact", and we rely on them to be just that. What is the nature of the difference between a mathematical knowledge, for example, 1 + 1 = 2, and a more general knowledge, suppose, Alice is dating Bob or universe started by big bang? Or do these knowledge all have something in common?

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Mathemathics is an ideal abstraction. It is in some sense discussible how real it is. :-) But it's based on perfect abstractions and can therefore reach a sort of perfect knowledge that things dealing with the real world can't reach. Because although 100*2000 = 2 000 000 in maths, if you take 2000 piles of hundred apples, a whole lot of them will be squished, so you end up with less than 2 million apples. The real world are often not quite as perfect and self-evident as maths is, so we are less sure of it. – Lennart Regebro Jun 19 '11 at 8:58
@LennartRegebro I may need to use that example. That's a really elegant way to capture that distinction! – Cort Ammon Sep 3 at 14:45

Although not a definitive answer to the complete depth of the question some light may be shed looking at the difference that Immanuel Kant makes between the two types of knowledge.

Mathematical knowledge is idealistic. It can be proven in a purely rational way, it is what Kant called a priori knowledge which means you have no need for the senses to come to a conclusion about it's being correct or not.

Knowledge of facts is empirical. It can only be proven through mediation from the senses, it is what Kant called a posteriori knowledge which means you need the senses for verification.

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From a foundational perspective: A considerable part of the philosophical activity of the 20th century has been devoted to this inquiry. Frege, Russel, Wittgenstein, the inquiries around Hilbert 2nd problem, formalization, and the following Goedel's theorems, and more followed that - Hilbert's first problem etc. .

So giving a brief answer is not only difficult but embarrassing - still, after suggesting studying the authors mentioned above (which I did a log time ago) one can try a brief answer, which is... that there is no difference, in that if the inquiry is pursued with sufficient depth and competence, some reliance on subjective intuitions is found in all cases, even 1+1=2.

From a non-foundational perspective: in this case one could analyze your sample questions as acts of language, socially contextual sentences, and more, and there find non-deterministic criteria to separate the questions in families with their internal rules. Here again there is simply too much litterature starting from (again) Wittgenstein - gets too wide a theme to get a single answer.

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If we consider mathematics the study of logical consequence, which seems reasonable (unfortunately I don't have time to write a paper on that right now) then mathematical knowledge is all knowledge derived from a collection of axioms, using a given system of logical consequence. In that case, a statement's truth value is only known in a given context of axioms and logical consequences used.

On the other hand, knowledge overall is far more vague. It is all information which we hold to be true, regardless of the method through which we obtain that truth value. As @ju5tu5 suggested, much of this knowledge is obtained empirically, but even then it does not have to be. It can be something at wihch we arrived simply because the conclusion satisfies us. A person may "know" that there is a good, because it provides them a great deal of subjective sanctification.

So I suppose in summary, general knowledge is the total collection of statements which we hold to be true, while mathematical knowledge is the collection of statements which we hold to be true based on logical consequence.

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