Definition talk:Inductive Argument
This definition is so vague as to be all but unusable in its current form. And as for the citation, I got my GEB out and read through Chapter VII (basically an explanation as to how propositional logic works) and could not find the definition of (or even a reference to) an "inductive argument". Also, it needs to be compared to what a mathematician usually understands an inductive argument to be. Basically, I looked at this page and couldn't make sense of it.
Suggestions for improvement:
- a) Give an example of an "inductive argument" so as to provide a hook onto which to hang the definition.
- b) Specify not only the chapter, but the section within that chapter (it's divided neatly into subsections of one page or less) so as to guide the reader into where in the chapter you are looking)
- c) Provide an instance of proof by mathematical induction so the reader can see what the differences are between that and this. --prime mover 16:49, 26 December 2011 (CST)
- The main part of it was from Salmon, I was just using GEB as a motivation as to "what the big deal is". I was worried that "types of inductive reasoning" should be on their own page, but I think you're right that I should add them here. I'll do so. --GFauxPas 16:59, 26 December 2011 (CST)
- After thinking about this for a while I realize that in order to make this concept cleaner I would be better off making "inductive arguments" a category instead of a page, because really it encompasses several types of arguments, each of which can be more clearly addressed individually. I have copied all the data for myself to work on later and chop up into more easily digested pieces: please delete the page. Or does anyone think a definition page would be better than a category? --GFauxPas 10:33, 27 December 2011 (CST)
- I don't know. How relevant to mathematics is an "inductive argument" anyway? I understand that in applied maths, physics and stats it can have its use, but then I also understand that in these contexts it's a standard technique to quantify your confidence percentage, and there's little room for the kind of "guesswork" that an inductive argument consists of. What's your thoughts on that? --prime mover 14:33, 27 December 2011 (CST)
- That's precisely where it is used in math, in quantifying confidence in applied math and stats. Inductive arguments aren't guesswork, they just cant guarantee truth preserving-ness, and that's a risk statisticians are willing to take, right? I confess I haven't taken statistics for around 8 years. --GFauxPas 15:12, 27 December 2011 (CST)
... Excuse me while I think out loud again, I'm going to have to run this one by my brain cells. Apologies if I ramble.
From my understanding, it's only a risk statisticians are willing to take when they can quantify how much risk. I understand that laws of physics are inductive (in this sense): "If I drop a hammer on a positive-gravity planet I do not need to look at the hammer to know that it has fallen," said Spock, "logic tells me ..." except it is not logic that tells him, it is his inductive interpretation of his experience that "On a +g planet, hammers fall." He only knows this because "that's what's always happened" - and ultimately, that's all you can deduce from physics. We know the "laws of physics" because we have studied them empirically. Thus, the "laws of physics" as we have started to introduce them on this site (we've only got Newton's 3 at the moment) are distillations of generalisations of observations and measurements.
Now mathematics is different. You can not deduce that "because the first billion zeroes of the zeta function are on the critical line, they all must be", or "because we've gone up to an exponent of 20 million or so we can just assume that the last 6 questionable numbers in the Sierpinski problem are etc. etc. etc." Statistics doesn't come into it at this level, and anything else is an unproved conjecture.
So, where does statistics come into it? When you're counting / measuring an everyday observation of mundane things (height of trees, length of waistlines, blablabla), you can make a guess at what the pattern is that links the measurements together, but I contend that this is not mathematics. It turns into mathematics when you have assembled your model, and are about to do the mathematical manipulations to calculate the mean, variance, skew, kurtosis, what-all. And at this stage an inductive argument is what is used to build a model. But that's about as far as it goes.
I gather that the main thrust of Salmon's work is to make that bridge between the real world of observations and the mathematical world of the constructed statistical model, along with a considerable amount of logic, taken from the context of semantic interpretation rather than from mathematical analysis. And to that extent, it's arguable as to whether it is actually mathematics. --prime mover 15:51, 27 December 2011 (CST)