Hume denies demonstrative reasoning, the ability to connect ideas to predict how things will work in the future. This basically amounts to the denial of our knowledge of induction. He denies this because our belief that the future will resemble the past is in fact derived from an instance of the same principle: we believe the future will resemble the past because the past futures have always resembled past pasts.
I’m wondering, though, how this applies to things like maths. Mathematical formulas seem to be abstractions from experience, but are formulated so definitely that they do not fall prey to accusations of question begging. And at the same time, they are applicable to experience, albeit in abstract ways.
I tend to agree for the most part that induction is flawed, but I don’t find that to a problem. No one claims guarantees from induction, only functionality. There is not certainity to be gained, that’s easily demonstrable, but that doesn’t mean that induction is not practically applicable, nor even that it is not useful in philosophical considerations, such as ethics.
This is Imp’s whole area but basically, as I understand it, Hume talks of maths as an analytical, metaphysical product of language that has no necessary connection (Imp would say no connection whatsoever but we know he doesn’t exactly mean that) with the empirical world of moments in flux. As such a mathematical statement like ‘2+5=7’ is true simply because we’ve chosen to define the terms in that way. It doesn’t mean that every time we have 2 and 5 grapefruits that we’ll get 7 grapefruits because that commits the fallacy of induction.
Well… if an event “always” turn out the same way is it not reasonable to recognize this and expect that the *PROBOBILITY of a particular outcome is high?
Strange but I should have an article going up on Symposia quite soon about Hume and Induction. Keep an eye out for it. As far as I would see it, I’d go with the view expressed already by someoneisatthedoor, in that Hume would see maths as a very different kind of thing, a ‘relation of ideas’ rather than a ‘matter of fact’ and as such is secure because it rests only on the relations of ideas, without presuming anything about the world.
Probability doesn’t really get us anywhere. In any sufficiently complex context, probability relies heavily on previous observation. Even in something as simple as the roll of a die, the probability calculation assumes that the sides all have equal chance of coming up, which is something we have experienced directly, or that we reason to be the case from experiences with physical objects.
But what about more abstract maths, like logic. While I can’t be assured that the logic gates on my computers CPU will function correctly, can’t I know that if they function correctly, they will yield a certain result? And if I can, don’t I then have some sort of insight derived from demomonstrative reasoning?
There’s the rub - we’ve never experienced ‘equal probability’ directly - it is a relation of ideas, not of objects (matters of fact). Using something we’ve constructed after abstracting from experience (namely the statement that ‘an even die is equally likely to land on any given side’) to then predict about the future is to leap from empirical experience to the metaphysical theory (in language, logical or mathematical or whatever) and then back from the theory to empirical experience without explaining what enables us to do that. Now this is the very basis for the Kantian synthetic a priori but as far as I’m aware no sound justification has ever been given for it.
Yes, because the definition of ‘the circuits functioning correctly’ is that they ‘yield a certain result’. No, because one never knows whether the computer is working because the circuits are functioning correctly or because the beret on the back of your chair is in the correct position. Unless one checks the circuits every single time one uses the computer. But no one does that, we simply take it on faith that it continues to work because it was engineered correctly, we’ve never actually proven this.
Induction is the process of making conclusions from instances and facts. A mathematical concept is not induced like a meaning is induced, as in the case of making causal predictions because it isn’t a function of instances and facts. They are intentional and therefore involve certain contingent inductions. “Facts” and “instances” aren’t always true, but this has nothing to do with the experience of phenomena and sensory data. There is no accountability in the experience of “two” grapefruit…one doesn’t have to call it “2.” One can’t go wrong in the experience of two grapefruit. They can spend forever defining what the grapefruit are and participate in an endless string of inductions, but that there are two is not a question.
Math cannot be written and remain what is was before it was symbolized. Now it becomes a representative, and it is no longer a matter of precise quantities but rather the empircial state of the symbol itself-- the actual typed number “three” is not an instance of the experience of “three” objects. The function of math is phenomenological in its purest form, and to become objective it must be suspended, as Husserl mentioned. I can no sooner provide for you a case of my comprehension of “three” than I could prove that I haven’t been tricked and “three” is really “four.” But I am very certain that I know the difference between two grapefruit and three grapefruit.
Math is not a paradigm. It is, if anything, a phenomenological manifestation of logic that happens through conscious negation. It doesn’t rely on symbolism or language or character to exist. To refute it is to use it.
By arguing against induction Hume commits induction, because if Hume cannot prove that a causal relationship exists between things or premises, it is quite possible that Hume’s conclusion doesn’t follow from his premise, in which case the same logic is present in the reasoning while the “facts” and “instances” are inductive probabilities. The same analytical function is active in an anti-inductive argument, the one which is sought to be proven does not work.
I don’t think induction is needed to comprehend quantities. Only qualities, things that make up facts and instances. These involve intentions and are transcendent to the sheer presence of things. The empirical gives itself to consciousness as an amount of data…what quality and meaning is derived from that must be somewhat autopoetic. It is “no longer” an objectivity with a simple shape, density, mass, size, etc., etc. It is now part of a plot and its participation is what it is intended to be an activity of. Qualities and values arise from this, and so does the induction fallacy. I suppose they go hand in hand.
Is that to say that induction must hold even for deductive arguments to work? While I am inclined to agree that deductive processes are learned under the assumption of induction, I think it is hard to show that they are not somehow independently justified. If induction is assumed, there is an argument to be made for the non-induction-dependence of deduction, in that it is fairly consistent across times and cultures and people. If it were really induction-dependent, such accord would not be expected.
I also like the idea that induction is justified by deduction. The line goes that if deduction is accepted, then there must be a regularity to the universe. Such a regularity would justify further the inductive step from multiple experiences of a connection between two phenomena, and the conclusion that the phenomena are somehow causally linked.
Just an idea…
The math instinct is a very basic instinct which can be observed in infants only a couple days old! I imagine that’s why philosophers over the ages have accorded it such a great degree of certainty and tried to alloy its strength with other areas of life. Because math is more instinctual than empirically learned, it is difficult to criticize it with the ‘problem of induction’ formula. It is easy for us to imagine how seeing many green leaves doesn’t imply that the next leaf we see will be green; it’s much harder for us to imagine one and one being anything other than two. I think this is more because the instinct is so strong than it is because math is ‘inherently’ more ‘certain’.
The book to read of course is Keith Devlin’s “The Math Instinct”.
