In answer to your question, the system of axioms that I'm using is the one that is usually used, stated and cited.
Well, I don't want to turn this into a thread about mathematical logic, but Peano arithmetic is one of the standard ways of defining the natural numbers — jkg20
, so I'd have to challenge you to provide a more "usually used, stated or cited" system of axioms for doing that.
In any case, you cannot define "1" this way (as you do):
Let "1" mean the multiplicative identity.
without already having defined multiplication (recursively) over the natural numbers, which means that the natural numbers need already to have been defined within your system
But in any case, you will not be able to infer 2+2=4 without all those axioms. MetaphysicsNow is right about that and you are wrong.
Please provide a link to a site where this "usually used stated and cited" axiomatic system is set out clearly, preferably by a mathematician.As I said, the arithmetical axiom system for the counting-numbers, with respect the the addition and multiplication operations, that typically and usually used, stated and cited is the one in which associativity is an axiom, not a theorem.
By "substantive truth", do you mean an alleged objective, fundamental, concrete "existence" for our physical world and its things and its stuff?
Now we are getting somewhere. Idealists and anti-realists more generally can (and have) made the distinction between substantive truth and logical truth, so no, one does not need to be a materialist in order for the distinction to make sense. You seem to be a fan of online philosophy encyclopedias
— jkg20
- look up "logical truth" and see how complex a notion it is...
...and how various philosophers have tried to distinguish it from substantive truth. As far as I can tell, every example of if-then fact that you introduce is an example of a logical truth
, but substantive truths are the ones that concern the empirical world (whether that world be independent of our coming to know it or not).
Now, if you want to argue that there is no genuine distinction between logical truth and substantive truth, that might be an interesting discussion to have.
Take a look at Quine's "Two Dogmas of Empiricism" for example.
Since we, as sentient beings, are in the empirical world, it is substantive truth that will have a bearing on whether or not we can be reincarnated, not logical truth.
If the truth of one proposition implies the truth of another proposition, that’s a state-of-affairs.
In other words, it’s a fact. — Michael Ossipoff
If the truth of one proposition implies the truth of another proposition, that’s a state-of-affairs.
In other words, it’s a fact.
If it is a fact at all it is a logical or semantic fact. Paris being the capital of France is a substantive fact. See the difference? — Janus
As I see it is on account of this difference that your 'argument' founders, or flounders.
Now, if you want to argue that there is no genuine distinction between logical truth and substantive truth, that might be an interesting discussion to have. — jkg20
Can you be more specific about what I said that suggests that?
You mean reincarnation? — Michael Ossipoff
Precisely. So, in order to get 2+2=4 out of this system of axioms for the real numbers, you need already a recursive definition of addition, otherwise you are stuck with 0 and 1 and the reals between them. Sure, you can add definitions (1+1)=2 , (2+1) =3, (3+1)=4... but either that is eliptical for defining addition recursively (as per Peano arithmetic for the natural numbers) or you need to supplement the axioms you link to with axioms for the existence of 2, 3 and 4. The system of arithmetic for the real number system trades off of the system of arithmetic for the natural number system and usually in order to prove that 2+2=4 you will need all the axioms of natural number arithmetic. In all cases you will need more than just the truth of the axiom of associativity for addition, so your conditional:But I'll add that the common arithmetical axiom-system that I refer to doesn't include a definition of multiiplication or addition, but merely mentions them as two operations, with respect to which the axioms for the number systems are stated.
is false, since (without all the required additional definitions and axioms) the antecedent could be true and the consequent false.IF the additive associative axiom is true, THEN 2 + 2 = 4
Other than what? Other than a system of relations? Relations relate things to other things, so the physical world - whatever else it is - certainly includes those things that stand in relations to each other, Metaphysics needs to address the nature of the things that stand in relations to each other as well as to the nature of the relations in which they do so stand.As I said, there's no evidence that our physical world is other than that.
As I said, there's no evidence that our physical world is other than that.
Other than what? Other than a system of relations? — jkg20
Relations relate things to other things
, so the physical world - whatever else it is - certainly includes those things that stand in relations to each other,
Metaphysics needs to address the nature of the things that stand in relations to each other as well as to the nature of the relations in which they do so stand.
Yes, reincarnation is implied if metaphysics is real.
You, the observing entity, the being is different from the body in the sense that body is material, it can't be aware of anything. It just works, you observe its work. And if you are beyond physical reality, death doesn't destroy you
.Just like you don't feel anything while asleep but "you're still there", you experience something else. But is it something like heaven/hell
or do you come back to earth?
Maybe it depends on the gravity of your mind. Maybe our minds weigh on a scale which decides the realm of existence you're in. Magic? Yes. Pure magic, not science at all.
But hey, you defy science in every sense of the word. Close your eyes and feel yourself for a moment. You'd observe you're not your thoughts, you're the one hearing those thoughts, feelings different sensations. You aren't visible, measurable or traceable. You are consciousness.
Experiencing, feeling, hearing itself is a proof of something beyond physical universe. In a pure material universe, chemical machines exist but consciousness can't.
I mean just think about it, you could build a robot but you can never make it aware of itself.
Trees, animals even babies, all have feelings and thoughts; however no one is feeling, thinking, they're just closed biological machines.
But, as I said, it's time to agree to disagree, and to discontinue this conversation.
Just admit that you have a scanty knowledge of number theory, — jkg20
This specific issue is not about agreement or disagreement, this is about you being wrong about what one can prove in mathematics given a set of axioms. — jkg20
this is about you being wrong about what one can prove in mathematics given a set of axioms — jkg20
there are no relations that relate Slitheytoves to Jaberwockeys either. There are logical relations that relatestatements about Slitheytoves to statements about Jaberwockeys.
However, there are distinct kinds of relata and relations which concern existent things
…and which your metaphysics remains utterly mute on
…because your metaphysics deals only with the logical relation between statements not the physical relations between existent things.
— jkg20Now, if you want to argue that there is no genuine distinction between logical truth and substantive truth, that might be an interesting discussion to have.
.There is certainly a logical distinction between logical truth and substantive truth.
.One might want to argue, as MO does, that there is no genuine metaphysical or ontological distinction, but then such a metaphysical argument could hardly be "non-controversial", as MO so mistakenly claims his metaphysics is.
.So, you obviously deny the indispensable logical distinction between semantic facts and substantive facts
., because it allows you to continue believing in your own sophistry.
.…your lack of cogent argument!
.Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not. However, logic does not deal with truth in the absolute sense, as for instance a metaphysician does.
.Logicians use formal languages to express the truths which they are concerned with, and as such there is only truth under some interpretation or truth within some logical system.
.
A logical truth (also called an analytic truth or a necessary truth) is a statement which is true in all possible worlds[46] or under all possible interpretations, as contrasted to a fact (also called a synthetic claim or a contingency) which is only true in this world as it has historically unfolded.
A proposition such as "If p and q, then p" is considered to be a logical truth because of the meaning of the symbols and words in it and not because of any fact of any particular world. They are such that they could not be untrue.
I do not need experimental evidence to know that I have experience. Indeed, the very idea of experimental evidence presupposes the idea that someone has already had experience of some kind. So there is at least one substantive non-if-then fact (to use your curious terminology): that there is experience.That there’s no experimental evidence that your experience is other than that.
So your metaphysics includes both non-if-then-facts and if-then-facts. — jkg20
For the former, their truth consists in some kind of relationship to the way things are
(and I leave open that the way things are might in part be determined by our ways of coming to find out about them, and thus allow for both materialistic, idealistic and pragmatist metaphysics).
For the latter, their truth consists in the logical relations between propositions used to express the non-if-then-facts.
So even here we seem to have introduced another non-if-then-fact: there are logical relations between propositions. Since your metaphysics depends on the existence of these relationships, how do you account for their existence? Is it just a brute non-if-then-fact?
But they're the metaphysical reality that underlies our pragmatic experential world and life
I make no claim for any reality or existence for the abstract if-then facts, or the infinitely-many systems of them.
"
"But they're the metaphysical reality that underlies our pragmatic experiential world and life"--Michael Ossipoff
Presumably by the "they" you mean your if-then facts. You seem to be contradicting yourself directly when you then go on and say.
"I make no claim for any reality or existence for the abstract if-then facts, or the infinitely-many systems of them.". — MetaphysicsNow
In that sentence, I didn't mean "reality" literally
:lol: :up:I'll leave you now to go ahead and play Lewis Carroll's Humpty Dumpty all by yourself, and I suggest @jkg20 do the same. .
Like one or two other people on this site, it seems that as soon as you are caught out in a contradiction, rather than rethink your position, you simply change the meanings of words. — MetaphysicsNow
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