I can't tell if you're serious or just trolling me out of boredom.
If I say that 5 = 5, is this something you would question? I'm just trying to figure out what on earth you are talking about.
I can't tell if you're serious or just trolling me out of boredom.
I am serious
Ok. Well for the sake of discussion, this is an interesting discussion!
I perfectly agree. I'm one who often takes the formalist stance. If someone says, "well how can .999... = 1 or pi have infinitely many digits with no repeating block or how can irrational numbers exist when they don't exist in nature," I just say: I don't care! You don't ask is chess is real. Math is a formal game. Stop asking me for it to mean anything. It's fun to play and keeps me out of trouble. Go harass the chess players and ask them if the knight "really" moves that way. You see the nonsense of expecting math to "mean" anything. Math is a formal game and Hilbert agrees!
It rather depends upon what one expects a philosophy of maths to be. Certainly maths started from the empirical and the real but this does not mean that this is the best way to incorporate it into the body of a philosophy. (People used to think that constant force results in constant velocity.. as per pro pushing a cart.. but physics has progressed beyond this model).wtf wrote: ↑Sun Sep 17, 2017 1:07 amOk. Well for the sake of discussion, this is an interesting discussion!
I was put on to the law of identity on this very forum, perhaps a year or more ago. Someone posed the following subtle question.
We know that in formal set theory, we have the axiom of extensionality. This axiom tells us when two sets are equal. Two sets are equal when they have exactly the same elements.
So the set {1,2,3} is the same as the set {3, 2, 1} because they have the same elements. Order doesn't matter. But they are both different than the set {1, 2, 3, 4} because that set contains an element, namely 4, that's not present in {1, 2, 3}. So those are not the same set.
Now the question is ... before I have formally defined equality of sets ... how do I know when two elements are the "same"? That is, if I claim that {5} = {5} because both sets contain exactly the same elements, namely the single element 5 ... how do I know the 5 on the left is the same as the 5 on the right?
I confess this question gave me quite a bit of trouble. Especially because it was posed by a poster who was, let's say, a bit cranky in his presentation, yet seemed to have a lot of insight into things. One often finds that among alternative viewpoints. Being crazy and being interesting are not mutually exclusive. On the contrary. It's sanity that's boring.
So I mulled this question over, and asked myself how we know that 5 = 5 before we have defined 5 as a particular set, as is done in the development of the natural numbers within set theory. In fact 5 is nothing more than the set {0 ,1, 2, 3, 4}. And therefore 5 = 5 by the axiom of extensionality.
But what is 5 BEFORE we have so modeled it within set theory?
The only answer I can come up with is that this is an application of the law of identity from logic. This is what I meant earlier when I said that the law of identity is logically prior to math. You objected to that. What I meant was that the law of identity is logically prioer to set theory. Whether you take set theory as a reasonable proxy for math is a question of philosophy of course!
So that's where I'm coming from. In math we start by assuming a thing is equal to itself. Then we stipulate the axiom of extensionality, define 5 as a set, and prove that 5 = 5 within set theory.
But before set theory, before any formal development of math, is the principle of identity. A thing is equal to itself. And yes I fully agree that this is something we must be explicit about once in a while. The law of identity is assumed. It's hard to imagine what kind of formal reasoning we could do if we were not allowed to assume that 5 = 5.
I perfectly agree. I'm one who often takes the formalist stance. If someone says, "well how can .999... = 1 or pi have infinitely many digits with no repeating block or how can irrational numbers exist when they don't exist in nature," I just say: I don't care! You don't ask is chess is real. Math is a formal game. Stop asking me for it to mean anything. It's fun to play and keeps me out of trouble. Go harass the chess players and ask them if the knight "really" moves that way. You see the nonsense of expecting math to "mean" anything. Math is a formal game and Hilbert agrees!
But then on the OTHER hand, of course math is derived from our experience in the world. Our experience of space and time and quantity and measure. And the study of formal math is inextricably bound up with our knowledge of the world. No physicist thinks it's all a formal game, they're studying the real world and they use math. Physicists mock mathematicians for their abstract formalism. Math is about the world. Ask any physicist!
So I see both sides of this.
What I DON'T see is trying to claim that 5 = 5 is questionable because you don't believe the law of identity. That is not a sensible thing to be arguing in my opinion.
But if all you want to do is get me to agree that the law of identity is an assumption, and you can't step in the same river twice, of course I agree with you.
The assumption that "a thing is equal to itself" is itself a particular worldview. Western rationality and all that. A perfectly sensible alternative to model the real world is: A thing is never equal to itself.
I have no problem with that. I rather think it's probably true about the world we live in. Western rationality is a point of view. It's useful. It's logical. It's fun. But is it necessarily at the heart of the universe? No, I rather doubt that.
I just don't see how we could do math that way. In math you have to start with the law of identity.
What do you think?
This is a strawman argument. Who is the entity that has an expectation that math is true about the world? Certainly nobody who's studied this history of science between the 1840's when non-Euclidean geometry was invented, and the early 1900's when Einstein showed that non-Euclidean geometry was actually the true geometry of the world.
Then we'd be back a few posts where I said that I myself am usually a formalist, and Hilbert's on board too. So I don't understand why you're addressing me as if I'm disagreeing with you on this point when in fact I agreed with this point several posts ago. I'm puzzled by your trying to explain to me the formalist position.
Now here you are wrong. If math is a formal game, then we definitely need to know what the rules are. Those are the foundations. If we're playing chess we need to know how the pieces move. If we're doing math we need to know what symbolic manipulations are needed. Formalists MUST have a foundation.
Really? Why not? Wouldn't we want to know that our rules can determine what is and isn't a legal position in our game? To know what's provable and what's not? Gödel showed that any formal system strong enough to model the natural numbers must be necessarily incomplete. That was a disappointment to what was at the time a requirement. Why would you lay down a set of rules then say we don't care if they completely determine the game?
And he showed that no such system can prove its own consistency. So if consistency is your only criterion, you don't have any criterion at all. We can not prove within set theory that set theory is consistent.
I have no disagreement with this. As long as you do acknowledge the "unreasonable effectiveness of mathematics" as the saying goes.
I think the laws of logic are called laws instead of hypotheses because they make intelligible communication possible. For us to be able to communicate meaningfully with each other we need to understand the basic structure of language and its correct use. This structure consists of definitions and rules governing the units of the language. This structure is called logic and the associated definitions and rules are embodied into laws. Since they make purposeful communication possible,i.e. without them there would be no possibility for intelligible communication, therefore they are called laws instead of just hypotheses.A_Seagull wrote: ↑Sat Sep 16, 2017 9:11 amSuch 'laws of logic' are not laws at all. At best they are hypotheses. It may be a basic principle of logic but it is not 'prior' to maths. Pure maths makes no use of such vague statements.wtf wrote: ↑Sat Sep 16, 2017 12:13 amThe law of identity is a basic principle of logic. It's logically prior to math. It's not an axiom of any system of math. "A thing is equal to itself" is a principle of reasoning independent of math.
There is an interesting entry on the Internet Encyclopedia of Philosophy (IEP) on this statement of Heraclitus. From IEP:
Daniel Graham wrote:Heraclitus does, to be sure, make paradoxical statements, but his views are no more self-contradictory than are the paradoxical claims of Socrates. They are, presumably, meant to wake us up from our dogmatic slumbers.
Heraclitus statement can also be dealt with from a Kantian perspective. This is interesting and needs to be considered in detail because it directly deals with the law of identity. Please, if you would allow me, here goes.
This is blatantly false.
The point about this quote is that the noumena changes, the phenomena changes; it is only the labels that remains the same.
Please, may I know from where and by what means did you come to know that Dolphins do not understand the structure of their own language and yet they are able to communicate meaningfully with that language?
Unfortunately, not everyone is laughing on this matter, for example the students who have failed either their TOEFL, IELTS or SAT exams are not laughing on this issue!
Respectfully my friend, one need not go as far as the distinction Kant made between 'noumena' and 'phenomena' for the discussion about the law of identity. However in Kantian philosophy, i.e. for Kant the 'noumena' is inaccessible/unknowable to him, all that he knows and can talk about is 'phenomena'. Kant was a specific type of idealist, he called himself a transcendental idealist. The particular flavor of Kant's idealism is not important here though, it only need to be mentioned that he did not believe himself to be able to access/know the things as they are in themselves. This point of view can be contrasted with the realist perspective, and from the latter perspective one believes that one has access to things as they are in themselves. For the latter view, it would imply that the 'noumena' and 'phenomena' distinction is superfluous.
This is essentially the same thing that Kant says in the Critique of pure reason, and in the case of Reid we need not even mention the 'noumena' and 'phenomena' distinction as he was a realist. So the argument in my previous post could have been appended after this quotation of Reid, instead of Kant.Reid wrote:The conviction which every man has of his identity, as far back as his memory reaches, needs no aid of philosophy to strengthen it, and no philosophy can weaken it, without first producing some degree of insanity.
(...)
We may observe, first of all, that this conviction is indispensably necessary to all exercise of reason. The operations of reason, whether in action or in speculation, are made up of successive parts. The antecedent are the foundation of the consequent, and without the conviction that the antecedent have been seen or done by me, I could have no reason to proceed to the consequent, in any speculation, or in any active project whatever.
(...)
I see evidently that identity supposes an uninterrupted continuance of existence. That which has ceased to exist, cannot be the same with that which afterwards begins to exist; for this would be to suppose a being to exist after it ceased to exist, and to have had existence before it was produced, which are manifest contradictions. Continued uninterrupted existence is therefore necessarily implied in identity.
(...)
When a man loses his estate, his health, his strength, he is still the same person, and has lost nothing of his personality. If he has a leg or an arm cut off, he is the same person he was before. The amputed member is not part of his person, otherwise it would have a right to a part of his estate, and be liable for a part of his engagements: it would be entitled to a share of his merit and demerit, which is manifestly absurd. A person is something indivisible, and is what Liebnitz callas a monad.
My personal identity, therefore, implies the continued existence of that indivisible thing which I call myself. Whatever this self may be, it is something which thinks, and deliberates, and resolves, and acts, and suffers. I am not thought, I am not action, I am not feeling; I am something that thinks, and acts, and suffers. My thoughts, and actions, and feelings, change every moment; they have no continued, but a successive existence; but that self or I, to which they belong, is permanent, and has the same relation to all succeeding thoughts, actions, and feelings, which I call mine.
[Essays on the intellectual Powers of Man, Thomas Reid, 1785]
In the above quotation, Reid is saying that the law of identity is something that he find himself to not be able to express in a proposition because it is "too simple a notion". After endorsing the law of identity, this would be exactly the position of Wittgenstein in the Tractatus more that a century later.Reid wrote:Identity in general, I take to be a relation between a thing which is known to exist at one time, and a thing which is known to have existed at another time, If you ask me whether they are one and the same, or different things, every man of common sense understands the meaning of your question perfectly. Whence we may infer with certainty, that every man of common sense has a clear and distinct notion of identity.
If you ask a definition of identity, I confess I can give none; it is too simple a notion to admit of logical definition: I can say it is a relation, but I cannot find words to express the specific difference between this and other relations, though I am in no danger of confounding it with any other. I can say that diversity is a contrary relation, which every man easily distinguished in his conception from identity and diversity.
What Reid expressed as "too simple a notion to admit of a logical definition", Wittgenstein would express as "that which cannot be said, but is shown in language". So a general conclusion on this subject could be that irrespective of one's religious beliefs and philosophical standpoint, the law of identity is not a subject of disagreement for the philosophers, or at least for the majority of them.Wiitgenstein wrote:Identity of object I express by the identity of sign, and not by using a sign for identity. Difference of objects I express by difference of signs. [Tractatus 5.53]
Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all. [Tractatus 5.5303]
The identity-sign, therefore, is not an essential constituent of conceptual notation. [5.533 Tractatus]
And now we see that in a correct conceptual notation pseudo-propositions like 'a=a', "a=b.b=c.⊃ a=c', '(x).x=x', '∃x(x)=a', etc. cannot even be written down. [Tractatus 5.534]
On the quoted statement above, someone might disagree with you on the basis of Heraclitus statement. On the same ground that the river would be changing according to Heraclitus, the labels too would not be exempted but would have a similar fate! A label is commonly referred to as a name, and in the case of the statement of Heraclitus, an example of a label would be the sequence of characters "river". Now, these labels or names can either be written on paper or displayed digitally on a computer screen or be a spoken word. As for the spoken word and the digital display, they are waves; sound waves and light waves respectively. And waves are changing like the river is flowing. In the case of the word written on paper, the label consists of atoms/molecules in either a solid or liquid state, depending on whether the ink has dried or not. In either cases, the atoms or molecules are moving or vibrating at least. Furthermore, the electrons within the atoms are moving at high speeds, and quantum physics even considers electrons as waves also. In a nutshell, if Heraclitus was right then nothing at all would be exempt from the fate of the river, not even the labels. But if Heraclitus was right, he would not be able to utter that statement itself.
That is false.
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