What makes mathematics certain knowledge?

The former is easy to prove with empirical evidence, but the latter is probably impossible to prove. It is the major issue of self-identity.

There is no such thing as knowledge in reality. Mathematics is just useful. What I assert is not knowledge. This is not problematic.

One could have an epistemology based upon 'use value' or 'pragmatism'.

That doesn't contradict my statement. Indeed, epistemology is, in my view, a branch of normative philosophy. When people say "I know XYZ" they are merely making subtle judgements about the degree of epistemic warrant required to assert a claim, and so forth. These are value judgements, value statements,and mostly normative. One can have "good" evidence. One can have "good reason" to believe. But not "knowledge". I think normative philosophy better helps us understand epistemic, moral, and aesthetic questions.

The pragmatist's understanding of "knowledge" goes against the standard conception of "knowledge". I don't think any fallible statement can be considered to be "knowledge". Indeed, many beliefs were, in the past, useful to have, but I wouldn't call them "knowledge". And our considered "knowledge" is constantly being revised due to the scientific method. Indeed, the operation of such a method suggests that our current understanding of the world is open to revision. Can we say that we "know" XYZ if it is open to revision? Can we say that we "know" that it's not open to revision, if that statement is itself possible to be revised?

Don't get me wrong - I have huge sympathy with philosophers who favour pragmatism in epistemology, but I don't think they're working with "knowledge". I'm sceptical about the existence of "knowledge".

And unlike Descartes, I don't think there is an example of true, justified, undoubted, believed knowledge that escapes Gettier and other concerns (e.g. Agrippa's Trilemma), never mind building up to any greater knowledgeable landscape.

Pragmatic and thus acknowledged 'fallible' knowledge is still a form of knowledge though, it just moves the question to how do we know something is 'less worse' knowledge than something else, I suppose this is the question of verisimilitude ?

An epistemology which acknowledges that all knowledge is fallible has trouble accounting for itself though, it seem either paradoxical or it is forced to give itself an elevated position outside all human contexts.

The idea that we have "better" knowledge and "worse" knowledge seems extremely counter-intuitive to me. It seems that you're using "knowledge" incorrectly. To say that "I know" that XYZ obtains yet conceive that "XYZ" might not obtain seems to me that you're using the term incorrectly - most people believe "knowledge" to be infallible (you either know it or you don't, there's no grey area for doubt). But if you want to use the word "know" instead of "think" or "believe" or "believe strongly", then fine. It's just a cosmetic, superficial change to the language you're using. I think it serves to obscure discussions, but you can define "knowledge" in that way if you want.

There is nothing paradoxical by just assuming that people use the term "knowledge" functionally (and inaccurately). It serves a useful function to say that I "know" that gravity causes apples to fall. It doesn't mean I literally know that we won't disprove that gravity exists (and that laws of nature will remain consistent and so we can use inductive reasoning), it just means that believing in it provides sense in a world where we need stable principles on which to act. I don't know that my sandwich will be in the fridge when I come back to my house, but I have good reasons to believe that it will be (which aren't countered by good reasons to assume that it won't be there), and I will act as if I won't need to pay for a meal tonight, assuming my sandwich hasn't been stolen. Likewise, with gravity. As creatures who need to act, who have evolved by observing patterns, anchoring stable principles to use which are helpful, and so forth, it makes sense that language such as "knowledge" enters our discourse, but it's not infallible (and not really what most people consider the be knowledge).

(It's worth saying that I don't know this view to be 'true'', whatever that means. I just see reason to favour this view because, in the absence of good objections, it seems to best coherently explain how discourse works; it is open to revision).

Fair point. I was using knowledge in a descriptive sense rather than a normative one I suppose, perhaps I should call them knowledge-claims? If we 'believe' that all knowledge-claims are fallible, then we need a way to decide if one knowledge-claim (say a proposition) is 'better' than another since we can't talk about knowledge-claims ever being absolutely infallible and 'true' as their legitimator (except for tautologies and maths etc). There are now degrees of knowledge rather than either/or.

Your epistemology is a kind of meta-knowledge, you are saying you 'know' this and that about 'we need to observe and perceive patterns etc', 'we need stable principles etc to act', that's a claim to absolute knowledge, which gives you a warrant to say that all other knowledge-claims are fallible or useful or whatever. It begs the question of how you know that...what is your warrant to believe this meta-knowledge-claim is true knowledge? What's it grounded on?

I'm not sure it explains much at all, it just mystifies it really in some human nature blackbox.

Melancholy, how much maths have you studied?

I don't think you got my point, and I don't think I get your point here. Again, I think to speak of "degrees of knowledge" is a nonsensical idea. Like "degrees of infallibility".


Again, I don't think you've understood my posts. I do not make claims to absolute knowledge. Indeed, I maintain that knowledge, as typically understood, does not exist. This talk of "meta-knowledge" complicates and confuses the point I'm making.


I don't think there's anything mystifying about it. B'ah well.


I'm wondering why this is being asked - do you have a contribution?

I'm wondering why this is being asked - do you have a contribution?

I'm a bit confused now since you make the absolute knowledge-claim that absolute knowledge does not exist...which as I said before is either paradoxical or must try and locate itself on some Archimedean stand point.

Even if we accept your claim that absolute knowledge does not exist, you must have some criteria for what constitutes knowledge, some warrant for it? If not then knowledge leaks into opinion and ignorance, and all is one. Unless you have some way to resolve the absolutist-relativist antimony.

...yes? Why else would I ask?

You're tantalising me now - I think you should make your point. As far as I see it, your question was irrelevant - you either have a problem with the argument or you don't. You should make a contribution and stop delaying; if you have a point, please say it. I have no idea why else you would ask, but equally I have no idea why you're not being forthright with your opinion on the matter.

I said explicitly that it is not an absolute knowledge claim, hence no paradox reveals itself.

I do have heuristic methods for believing what to believe to be true - a faith in gravity hasn't let me down yet, mathematics "truths" usefully allow me to conduct work and achieve a useful answer, certain coherent sets of answers help me to make sense of the world. I don't see why you insist on using the word "knowledge" in trying to describe my position, never mind "absolute" knowledge. I develop reasons for choosing some beliefs as more epistemically warranted - as do most humans, naturally.

The reason I asked is because the way you describe mathematics is very skewed, but your views would be understandable if you've not studied any mathematics post A-Levels (or equivalently, have no idea what mathematians actually care about when e.g. solving an equation).

I won't pretend to have any formal philosophical training, but from what I've read of philosophical logic or a philosophers view of say set theory, every author urges its readers to at least dabble in the way mathematicians 'do it'. This seems quite sensible to me - how do you seek to rationalise (or even describe) something which you haven't experienced yourself?

So I ask you once again; how much maths have you actually studied?

What is it then?

Do you think such a thing as knowledge exists or not then, it seems if you don't then you have no business communicating to me that you 'know' that knowledge doesn't exist...or at least I have no business 'believing' your claim is true since you deny that universal epistemological warrants exist? An epistemology based on personal institution in interesting I suppose but also a mystification and is solipsistic.

If maths just exists in the mind why is it so hard to learn?