We may divide truths into two kinds: receptive and consequential. — Zuhair
The fellowship of those consequences from the two input rules is a kind of TRUTH. This is a consequential fact. — Zuhair
Here I'm stating a similar stance that is: Mathematics is the study of consequential truth. — Zuhair
In epistemology, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that world. Correspondence theories claim that true beliefs and true statements correspond to the actual state of affairs. This type of theory attempts to posit a relationship between thoughts or statements on one hand, and things or facts on the other. — alcontali
imagine that you have an idea of a structure you want to capture the behaviour of, and you have the behaviour — fdrake
If ever you've tried to axiomatise a structure you'd see that there's a reciprocality between the structure's concept and its mathematical definition. — fdrake
Any form of reflection on the "world" or "states of affairs" concerns facts, and facts are always historical. Are you prepared to assert that all facts are true - without some weird question-begging qualification? — tim wood
My own definition of truth, fwiw, is that "truth" is an abstract term that simply means that the proposition in question complies with an appropriate standard in being true, while being entirely agnostic as to what that standard is. — tim wood
Well, polymorphism is obviously permissible as long as it does not lead to fundamentally ambiguous situations — alcontali
If by polymorphism you mean that truth can take different forms, I disagree, because on my definition, "truth" is defined to accommodate all forms, as qualified, because that is the best it can do. — tim wood
And I cannot think of a fundamentally ambiguous situation. — tim wood
I in opposition to your terminology prefer to use the term truth to denote another context which is quite different from the "correspondence with reality" context, and that context is what I've labeled as "consequential truth", you are free not to call it truth, you may term it as "consequentiality", or "consequential processing" which are fair enough. — Zuhair
Are you trying to program a structure by attaching invariants? — alcontali
Saying that it is a "group" automatically attaches a set of invariants. If you add enough invariants to the structure, i.e. you may use up all your degrees of freedom, then indeed, at some point there will only be one candidate definition that fits the bill. It could, for example, leave only one K possible. You could obviously also over-specify and propose the structure of something that cannot possibly exist. — alcontali
We find out what axiomatic systems give what conclusions, but notice that the conclusions that we desire are the motivating feature in this diagram. — fdrake
Pure formalism just gives you the black arrows, it does not give the sense of mathematical progress through the articulation and codification of ideas, just the dynamics of symbols, as if the ideas motivating them were completely irrelevant. Another way of putting it: formalism is just what we invent to get to where we need to go. — fdrake
CT truth and logical truth (LT) have also nothing to do with each other. For example, in "var b = true" the variable b does not correspond to anything in the real, physical world, but we cannot deny that it is logically true, if only, because we defined it to be.
Provability (PR) and logical truth (LT) have also nothing to do with each other. For example, in first his incompleteness theorem, Gödel encodes a statement that is logically true but not provable
So mathematics is about studying rule following games. I call them games because the choice of the primary rules is IMMATERIAL, we can even call them ARBITRARY, the most important is to harvest consequential truths in those games. The reality of the games, i.e. the stance of its primary rules and consequential outcomes from reality, is not relevant to mathematics itself, it is however relevant to its application, but not to mathematics per se.
analytic truth truth by virtue of the meaning of the words of a statement, synthetic needs meaning and correspondence with reality as well. Here with this terminology I'm speaking about rule following which can even be of strings of empty symbols, so meaning is not involved here, however synthetic seems to be overlapping with receptive truths. I think that analytic is "meaningful consequential truths", so I think the term "consequential truth" is weaker than analytic truth, although of course you can object to this by holding that consequential truth is a kind of non-meaningful analytic truth or by saying that rule fellowship is a kind of meaning, you can call it meaning by having a role in following a rule, if so then we can subsume consequential into analytic. The new things is that KANT was saying that mathematics is apriori synthetic. Which this philosophy doesn't agree with. I more agree with Hume that mathematics is purely analytic nothing else. — Zuhair
T+con(N) would rule out any statements like " this is not provable " and so on. Other than that, what will be a consequential truth of T itself and how does it differ from consequential truth of T+con(N) as a S that is a logical truth in T should also be a logical truth in T+con(N) .In other words S is provable in T+Con(N), that's why its said to be a logical truth, in reality it means that it is a consequential truth from the rule following game T+Con(N), notice that it is not a consequential truth of T itself. "Logical truth" is provability in some system
So mathematics is about studying rule following games. — Zuhair
I've already explicitly stated in my head post that I don't agree with formalism, — Zuhair
which can involve meaning as part of those rules, — Zuhair
Again the reality of those concepts are not the job of mathematics, it is the consequential load in those rule following games that is mathematics. — Zuhair
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.