## Infinity

• 2.3k
It was commented "I wonder what "nicknamed" would imply in supposed rigorous logic."

The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas.

Moreover, the nicknaming (my word) I mentioned is not so much about the logic but rather about defined symbols in a theory such as set theory.

Set theory, as formalized, uses only formal symbols, not natural language words. A formal proof is not allowed to use connotations, associations or any of the suggested notions that natural language words have. However, for everyday communication of proofs among mathematicians and students it is unwieldy to recite exactly each formal symbol in the formulas that are sequenced for a proof. Moreover, it aids picturing the content of the theory to informally use words. I call that 'nicknaming'. For example, in set theory in all formality, there is no constant term 'the empty set'; instead there is a 1-place operation symbol, a pure symbol, with a purely formal definition. Moreover, as I've mentioned, the adjective 'is a set' or even a formal predicate for 'is a set' are not even required as in formal theories such as ZFC. So, as concerns the formal theory, it doesn't matter whether or not one's personal notion of sets allows that there is one special set that has no members that is called 'the empty set' but rather, the theory has a formal theorem, such as:

E!xAy ~yex
thus a definition
0 = x <-> Ay ~yex

So, in that particular regard, we could just as well use the nickname "zee zempty set". It would not change the "structure" of the mathematics, which is the relationships of the definitions and theorems.

Note that my remarks about this are not necessarily a commitment to extreme formalism expressed as "mathematics is just a formal game of symbols". Rather, in this context, we may note that, no matter what framework or philosophy one has for understanding mathematics, at least we have the formalization, even just the fact of that formalization, as a component in our understanding - whether a fully self-contained and isolated component (i.e. extreme formalism) or as merely a point of reference and a rigorous constraint against handwaving.
• 3.5k

I agree. There was developing an interesting discussion on the law of identity and (non-ordered) sets. Or so it seems, I just glanced at it.
• 2.5k
So an hourglass changes its identity as each sand grain drops.

A moment of clarity.
• 2.3k
Within a different framework, say that of binary numeral system, 1 + 1 = 10

That's not an example of what I was talking about. I'm talking about general frameworks such as hold one's intuitions, perspective or philosophy, not matters such as variations in base numbering systems.

But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects.
— TonesInDeepFreeze

Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists?

No.

What does "it, the knight on a chess board, refer to?

When I wrote 'it', I was not referring to a particular piece of wood or particular array of pixels on a screen. I'm referring to the idea, the thing that players who are not even in each other's presence - thus not moving the same piece of wood or even seeing the same array of pixels - can still refer to as "the knight".

"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world.

But it doesn't. Again, one may argue that leading up to the formation of the concept, there were particular pieces of wood or ivory or whatever that were carved to resemble a knight and that were moved around on a two-colored board. But soon enough, we have the abstraction that can be referred to. Indeed, 'chess' itself can be defined mathematically without even mentioning particular characters such as 'knight', but rather only a purely mathematical construct. We could say there are four objects, called 'WKL', 'WKR', 'BKL', 'BKR' ("white" and "black" "knights" on "left" and "right") and the other "pieces", then define a C-sequence (sequence of "chess moves") to be a certain sequence of matrices with those objects associated with cells in a matrix and successive matrices having a property that the "pieces" are in different cells only according to certain allowed ordered pairs of cells ("moves"), etc. So you see that when I say 'it', I'm talking about an abstract object, even though attaining that abstraction required previous concrete or ostensive understanding.

Innatism

I am not opining whether or not the basic concepts 'is', 'exists', 'same', etc. are innate. Indeed, I have no ready argument that they are not first understood only ostensively. I'm only reporting that I don't know how I could arrive at successively more involved frameworks without them.

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts.
— TonesInDeepFreeze

But how can there be wisdom in the absence of facts.

I don't say that there can be.
• 2.3k
There was developing an interesting discussion on the law of identity and (non-ordered) sets.

More a painfully needed, though unsuccessful, intervention than a discussion.

The points are simple:

* In mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!"

{'The Maltese Falcon', 'Light In August', 'The Stranger'} = {'Light In August', 'The Stranger', 'The Maltese Falcon'}

{8, 5, 9} = {5, 9, 8}

And '=' reads fine whether 'equals', 'is identical with' or 'is'.

No law of identity is violated there.

/

Boss: Jake, tell me what is the set of items on our shipping clerk's desk?

Jake: It's the set whose members are a pen, a ruler, and a stapler.

Maria: But he also has another set on his desk! It's the set whose members are a ruler, a stapler, and a pen.

Boss and Jake: Wha?

Boss: Maria, take the rest of the day off. You're not quite with it lately.

/

{pen, ruler, stapler} = {ruler, stapler, pen}

Nobody says that the set of items on a desk is different depending on the order you list them.

On the other hand, mathematics does have ordered pairs and triples. For example:

<b c d> = <x y z> if and only if b=x, c=y and d=z.

With ordered tuples, yes, order does matter.
• 12.3k
The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas.

Since soundness requires true premises, and the logic you are talking about proceeds from axioms, which are not truth-apt, instead of from true propositions, how do you propose that it could be "proven that the logic is sound"?

mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc.

The point I made, is that the sense of "identity" you use here, is not consistent with the sense of "identity" used in the law of identity. So it doesn't really matter that you insist that "=" stands for "identity". Anyone can make up one's own personal sense of "identity" and have a symbol for it, state the axiom, and persuade others to use the axiom, and even create a whole "identity theory", but that doesn't make that sense of "identity" consistent with the law of identity.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!"

This is an excellent example of why your sense of "identity" is not consistent with the law of identity. By the law of identity, if the identified thing is "the books on your desk", then everything about that thing, including the order of the parts, must be precisely as the books on your desk, to satisfy the criteria of "identity". Stating an order other than what the books on your desk actually have, would not qualify as an identity statement, because that specific aspect, the order of the parts, would not be consistent with the thing's true identity.

No law of identity is violated there.

I have become fully aware that you are not at all familiar with the law of identity. Therefore your statements about the law of identity, and whether it is violated under specific conditions, I simply take as off-the-cuff remarks of a crackpot.

Nobody says that the set of items on a desk is different depending on the order you list them.

Anyone with any degree of common sense recognizes that the identity of the specified thing, "the items on a desk" includes the ordering of the mentioned items. If describing those items as a "set" means that the ordering of the mentioned items is no longer relevant, then you are obviously not talking about the "identity" of the specified thing, which is "the items on the desk". You are talking about something other than "identity" as defined by the law of identity.
• 3.5k
It's so easy for practitioners of the subject:

$2(x+5)\equiv 2x+10$ Identity

$2(x+5)=3$ Conditional
• 2.3k

Here's how I would put it:

2(x+5)= 2x+10
is understood to be implicitly universally quantified:
Ax 2(x+5) = 2x+10
and that is true

Then, by universal instantiation, we have:
2(x+5) = 2x+10
and that is true for every assignment of a value to the variable 'x'

2(x+5) = 3
is understood to be implicity existentially quantified:
Ex 2(x+5) = 3
and that is true

In high school algebra, we are asked to state the members of the "solution set", which is to say:
{x | 2(x+5) = 3} = {-7/2}
and that is true
• 2.3k
First order predicate logic may be formalized in two ways:

(1) With logical axioms and rules of inference. (Known as 'Hilbert style'.)
or
(2) With only rules of inference. (E.g., natural deduction.)

In either case, we have the soundness theorem, since the logical axioms are true in every model and the rules of inference are truth preserving.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

The law of identity is that a thing is identical with itself. Using '=' to stand for 'identical with' is not inconsistent with the law of identity. Indeed, the law of identity is stated:

Ax x=x

That is, for all x, x is identical with x. That is, for all x, x is x.

Not only is '=' as used in mathematics consistent with the law of identity, but the law of identity is itself an axiom of identity theory that is taken as part of the logic used for mathematics.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

"the ordering of the set" is meaningful only when there is only one ordering of the set (in this context, by 'ordering' we mean a strict linear ordering). Any set with at least two members has more than one ordering.

The set whose elements are all and only the members of the Beatles in 1965 has 4 members, and there are 24 orderings of that set.

In other words, there is no "the" ordering of the 4 membered set {x | x was a member of the Beatles in 1965} since that set has 24 orderings.

/

"crackpot" is said by the crank calling the pot cracked.
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