What makes mathematics certain knowledge?

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To the question of what is it: I have already answered that. When people say "I know XYZ", they usually just mean that they have reason to believe XYZ, often through inductive reasoning on experiences, or because believing XYZ has been useful/produced anticipated results in the past, or for some other reason; but it's not "knowledge" to the extent that it is not infallible (which I would assume is involved in most people's understanding of the word "knowledge"). I don't see how scepticism is to be avoided; and when you hear of how the word "knowledge" is used in real life, it seems closer to what I'm outlining.

Are we to believe that just because someone says "I know XYZ" that they have good reason to believe it is justified true belief?

However I'm open minded about radical epistemologies or some of the anti-naturalist ideas of say Wittgenstein trying to located 'knowledge' in language-in-use and 'seeing' how 'to know' is used in social practice.

I don't think 'to know' implies infallibility. One might say 'I know the bible', but not be able to recite every word for example, but by consent of one's peers one could be deemed to 'know' the bible still. Just as another example Popper's falsifyabilism implies such, since propositions that count as 'knowledge' are only ever unfalsified, and as such they are only not untrue, they can't claim to be 'absolutely true' ever.

Original post by Melancholy

I've already said, incredibly explicitly, that I do not believe that knowledge really exists. I have never said that I "know" that knowledge exists. However, it does not follow that you have no reason to "believe" that my argument is "true". That's a mistake in your argument.

I believe that epistemic warrants exist - they don't guarantee truth or knowledge, but there certainly are reasons to believe in things, and I'd probably call them epistemic warrants.

I can see you are struggling to avoid using the word knowledge here, but you talk about epistemic warrants so you are implicitly using it. I think you are mistaken in restricting the normative definition of 'knowledge' to what is known as absolutely true, and the skeptical position demands a more nuanced stance.

Anyway I said before something like how to you know absolutely that absolute knowledge doen't exists. There are antimonies in this knowledge-claim I think, or at least it can't account for itself.

Reply 61

shamika

Original post by Melancholy

Can you elaborate - how is it skewed?

You talk of mathematics as if it follows the empirical method (or at least, that's the impression I get from what you've written). There certainly are some who would suggest this is indeed the 'correct' model for mathematics, and many theoretical physics topics which are amenable to such an approach.

Pure mathematics (the topics which only mathematicians study because they are interesting in their own right, not because they are applicable) does not work in this way at all. A pure mathematician will spend many years trying to solve a problem for the simply joy of having learned something in the process. When the pure mathematician claims to have proved something, what he really means is 'to the best of my knowledge, I have come up with a series of logical deductions which are consistent with everything proved previously, from which the desired conclusion follows'.

Mathematicians (except those interested in foundations) do not actually ponder the certainty of their results anymore than say a Physicist. In particular, mathematicians are now more aware than ever that there is a distinct possibility of an underlying inconsistency in their proofs. They already know that a naive set theory is insufficient to demonstrate provability ever since Godel.

I've had to explore maths as part of certain philosophy modules (value, not philosophy of mathematics - in particular the parallels between the domain of discourse about mathematical concepts and moral concepts).

Interesting! I would argue that notions of morality are far too fluid to be constrained meaningfully by an axiomatic approach such as mathematics. That is, the mathematician in me would attempt to declare 'axioms of morality' from which you could deduce all other problems of morality. If possible, you'd be able to solve any moral dilemma by treating it as a mathematical proof :smile:

(I'm guessing that's not a parallel you actually considered in any seriousness.)

My housemate is a pure mathematician, my girlfriend is a statisticians; my own formal education is up to A-level, but I don't see how that is a barrier provided that you're familiar with, say, set theory - you'll find most philosophical papers that discuss parallels between, say, normative theories and mathematics will outline the information needed to make the paper's thesis intelligible.

Because a mathematician would argue that to be able to say much about the foundation of maths, you really need at least a couple of courses in set theory. I've had none. Besides, current mathematical thinking is that set theory is insufficient and category theory is now the current craze. My point is, the way a philosopher and a mathematician tackles the question is still a world apart. When a mathematician is studying category theory, he's really wondering 'can the study of the structure of the underlying mathematical object tell me something interesting about the objects I didn't already know?'.

And unless, you're questioning their mathematical training, I don't see how the question is relevant.

From the preface of Michael Potter's 'Set theory and its philosophy': "the philosophical literature contains far too many articles marred by elementary technical misunderstandings..."

(it would be remiss for me not to complete the sentence, for he has a warning for the mathematician turning to philosophy):

"...while mathematicians have often been tempted, especially in later life, to commit to print philosophical reflections which are either wholly vacuous or hopelessly incoherent."

I agree with both points entirely. I don't see how it is reasonable for someone to claim to have something interesting to say about the philosophy of mathematics without having some training in the basics in both.

Indeed, even for the foundational discusions in philosophy of maths (realism, irrealism, platonism, empiricism, conventionalism, and so forth) I doubt you'd need to know much other than a number line and the key operators in order to make some primitive statement on the way that those aspects of mathematics operate, perhaps geometry, meh.

But don't you see that all of those arguments are severely lacking, and cannot by themselves represent an adequate explanation for more than a few questions?

I would argue that the number line is a concept which is incredibly rich: mathematically, you can rigourously base the theory of irrational numbers, limits (and thus Analysis), measure and cardinality from the number line. It's not a simple concept! Mathematicians are still trying to prove very basic facts about numbers, hundreds (or even thousands) of years since a problem was first posed.

I understand that there is a key distinction between pure and applied maths, but hey ho.

The distinction is usually one of convention and the inter-relation between the two is hugely complex. Agreed however, that it is of tangential relevance to the topic at hand.

Mathematics itself cannot conclude whether its own content is true,

Really, I think this is the crux of the argument. When I was beginning my undergrad degree I thought this was absurdly false. Now, I'm not so sure. Mathematicians claiming to work in category theory and other related fields because they want to understand the foundations of maths are mostly kidding themselves. There are far too many competing theories for much progress to be made, unless someone remarkable makes a wholly unexpected discovery.

I actually don't know what to think at the moment. I'm not convinced that mathematics can't conclude its own content is true ('just set up a convincing foundational theory like what set theory was designed to do').

but I agree that some sort of familiarity with maths is necessary.

To summarise, I think where we disagree is that I think a lot of training is necessary, whereas you think something meaningful can be said with relatively little.

Furthermore, little philosophical training is, in my view, no barrier to discussing these questions - my primary training isn't really in philosophy. I'm mostly a history student.

Discuss yes, but say something interesting and insightful? Nope, don't think so. To think we could would be doing a disservice to anyone studying philosophy!

Reply 62

snozzle

Original post by Post121

Languages are clearly human mind constructs. And I don't think anyone can easily claim it's not difficult to be fluent in an arbitrary foreign language.

If something is a priori though, that is to say a kind of pre-knowledge in the mind then why would one need to learn it?

I think the point is that maths somehow 'exists' a priori in a supra-individual or inter-subjectively?

Reply 63

Post121

Original post by snozzle

If I imagine up a outer space monster in my mind, it clearly exists in my mind. But without concept of what's outer space, I wouldn't able to construct such monster. So would you argue the absence of knowledge of outer space upon my birth means that such monster can't exist in my mind?

A great way to illustrate maths is a human mind construct is realise our first coherent number system integers poorly explains the world.

Reply 64

Donald Duck

Mathematics is only absolutely true if we take a few axioms as true (stuff like 1 =/= 0).

From these axioms, all other mathematics can be deduced, so if these are true, the rest must be true as well.

The axioms are incredibly obviously true, but within mathematics, are not doubted.

Reply 65

Newton's Apples

Original post by Martyn*

“Mathematics would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.” - Friedrich Nietzsche.

That's possibly the most, and I mean, THE MOST retarded quote I have ever read. I laugh at its complete stupidity.

Reply 66

Negaduck

Why does the OP think that being overly aggressive makes him seem more intelligent?

Reply 67

~MiserableLogic~

Original post by Negaduck

Why does the OP think that being overly aggressive makes him seem more intelligent?

Why would I make such a faulty association? I am not aggressive and severely doubt my intelligence daily. :confused:

Reply 68

Aoide

Original post by Donald Duck

This pretty much covers it. Don't view it as language vs math, numbers are just another language used in situations where other languages wouldn't be adequate. Just like other languages it is a way of expressing concepts as symbols. All of our mathematical knowledge is deduced from a small number of rules and "facts". We know these are true because that is what we say they are. For example 1+1 =2 becase the definition of 2 is that it is double the size of 1. These are the basic rules of the langauge from which we have extrapolated everything else. Asking what makes mathematical rules true is like asking why the word for table is table. Maths is a human construct build to model a specific type of situation, just like any other language.

Reply 69

snozzle

Original post by Aoide

Yeah but what natural language has explicit rules which cover all instances of its use? Natural language is massively maleable.

Reply 70

TheJ0ker

Original post by zoe_bartlett

Mathematics is a social concept. It was not discovered, it was invented to help us better understand other subjects like science.

No, no it wasn't. It is not a human social concept either, all animals have a basic intuitive grasp of maths.

Reply 71

Bobifier

There seems to be a lot of confusion in this thread about a basic fact of Mathematical Philosophy that I would like to clarify. Godel's incompleteness theorems state that given an axiomatic system that can characterise arithmetic then there exist some statements that are true but cannot be proved to be true, and an example of one such statement is a statement of consistency of the system itself.

To put this in clearer terms, that means that in Mathematics there are things that are true but that we can't prove. It also means that if we prove that our system contains no contradictions then it contains a contradiction. This means that we cannot conclude that our own content is true (since if we do so then our content is false).

I think that a basic appreciation that these things have been proven beyond doubt mathematically will clear up a decent amount of the discussion going on.