From a couple of years ago, not involving fishfry. Probably there are others over the last couple of years. For simplicity, I didn't mention the quantification (in the meta-theory) over formulas:

x=x ... reflexivity
with
(x=y & Px) -> Py ... indiscernibility of identicals (aka substitutivity)

is a complete axiomatization of identity theory — TonesInDeepFreeze

From this thread, not addressed specifically to fishfry:

Identity theory is first order logic plus:

Axiom: Ax x=x

Axiom schema:
For all formulas P,
Axy((x=y & P(x)) -> P(y))

Semantics:

For every model M, for all terms T and S,
T = S
is true if and only if M assigns T and S to the same member of the universe. — TonesInDeepFreeze

The first mention that extensionality provides only a sufficient condition (which fishfry now regards as an epiphany for him), though not addressed specifically to fishfry, but there were at least a few others addressed specifically to him:

With identity theory, '=' is primitive and not defined, and the axiom of extensionality merely provides a sufficient basis for equality that is not in identity theory. Without identity theory, for a definition of '=' we need not just the axiom of extensionality but also the 'xez <-> yez' clause. — TonesInDeepFreeze

In this thread, not addressed specifically to fishfry.

Eventually, mathematical logic provided a formal first order identity theory:

Axiom. The law of identity.

Axiom schema. The indiscernibility of identicals. — TonesInDeepFreeze

In this thread. Comments on the subject, not addressed specifically to fishfry:

https://thephilosophyforum.com/discussion/comment/911857

In this thread, addressed specifically to fishfry:

identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals) — TonesInDeepFreeze

In this thread, addressed specifically to fishfry.

As I said much earlier in this thread, it is the first order theory axiomatized by:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For all formulas P(x):

Axy((P(x) & x = y) -> P(y)) (indiscernibility of identicals) — TonesInDeepFreeze

In this thread, fishfry himself quoting me:

The identity relation on a universe U is {<x x> | x e U}. Put informally, it's {<x y> | x is y}, which is {<x y> | x is identical with y}.

Identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals)
— TonesInDeepFreeze — fishfry

In this thread, addressed specifically to fishfry:

Start with these identity axioms:

Ax x=x (a thing is identical with itself)

and (roughly stated) for all formulas P(x):

Axy((P(x) & x=y) -> P(y) (if x is y, then whatever holds of x then holds of y, i.e. "the indiscernibility of identicals") — TonesInDeepFreeze

In this thread, addressed specifically to fishfry:

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals — TonesInDeepFreeze

In this thread, fishfry again quotes me stating the axioms:

And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals

— TonesInDeepFreeze — fishfry

Then, in this thread, fishfry denies the plain record of the posting, denying that I had never neve said what I mean by identity theory, even though he had not very long before quoted me stating the axioms:

I stated explicitly several times that that is what I mean by 'identity theory'.
— TonesInDeepFreeze

You never said that LOL! — fishfry

So I don't understand why fishfry would deny the plain record of the postings here.

Not only did I indeed state the axioms of identity theory several times, I stated them directly to fishfry, and he even quoted my statement of them.

Later he denied that I stated them, even though I had stated them not very many posts prior.

What explains fishfry's bizarreness?

Symbols need explanatory words to go with them. — fishfry

(1) The symbols I used are common. The formulas I gave are not complicated. If one knows merely basic symbolic logical notation, then one can read right from my formulas into English. For example:

AxEy yex

reads as

For all x, there exists a y such that y is a member of x.

For example:

(P(x) & x=y) -> P(y)

reads as

If P holds for x, and x equals y, then P holds for y.

And now that I've provided a one-stop list of the most common symbols, it's even easier.

(2) I did give lots of explanations in certain contexts.

(3) You complain about the length of my posts, but also say I should give more explanation. You can't have it both ways. And you're hypocritical since your own posts are often long, and often enough have not merely a few symbols.

Now explain this to me ONCE AND FOR ALL. Are we talking about pure math and set theory? Or are we talking about the physical world of time, space, energy, quantum fields, and bowling balls falling towards earth? — fishfry

I don't understand you. I gave you an example of how equivocation of "same" has a considerable effect. Of course it has no effect in "pure mathematics", because by definition "pure" mathematics maintains its purity, and the purity of its definitions. Pure mathematics is not applied, and therefore has no effect in relation to the physical world where "same" means something else.. We live in the physical world, our cares and concerns involve the world we live in, it is impossible that anything in the fantasy world of "pure mathematics" could actually concern us. This is known as the interaction problem of idealism. However, in reality we apply mathematics and this is where the concerns are.

You seem to misunderstand the issue completely. You appear to understand that there is a difference between the use of "same" by mathematicians (synonymous with equal), and the use of "same" in the law of identity (not synonymous with equal). You said that this difference has no bearing on anything you know or care about. The things included in the category of what you know and care about, are not limited to principles of pure mathematics, because you live and act in the physical world. The law of identity applies to things in the physical world which we live and act in.

So, to make myself clear, I do not claim that there is a problem with using "same" as synonymous with equal, within the conceptual structures of mathematics. The problem is in the application of mathematics, as inevitably it is applied, and this use of "same" is brought into the world of physical activity, and taken to be consistent with the use of "same" when referring to physical objects. That is where the problem occurs. Sophjists such as Tones enhance and deepen the problem by arguing that the use of "same" in mathematics(synonymous with equal) is consistent with the use of "same" in the law of identity (not synonymous with equal).

You can not have it both ways. — fishfry

This is exactly the issue, the reality of the situation is that we do have it both ways. There are two very distinct ways for understanding "same". You can dictate "you cannot have it both ways" all you want, but that's not consistent with reality, where we have both ways of using the term. If you think that we ought to reduce this to one, (insisting "we cannot have it both ways"), the two cannot be combined, or reduced to one, because they are fundamentally incompatible (despite what the head sophist claims). This means that we have to choose one or the other. If we choose the one from pure mathematics, then we have nothing left to understand the identity of a physical object in its temporal extension. If we choose the one from the law of identity, then we simply understand "equal" as distinct from "same", and the problem is solved. Obviously the latter makes the most sense, and doing this would support your imperative dictate: "You can not have it both ways."

No. You don't understand how math works, and you continually demostrate that. — fishfry

It is very clearly you are the one who does not understand how math "works". Math only works when it is applied. "Pure mathematics" does nothing, it does not "work", as math only works in application. You are only fooling yourself, with this idea that pure mathematics is completely removed from the physical world, the world of content, and it "works" within its own formal structures. That is the folly of formalism which I explained earlier. To avoid the interaction problem of Platonist idealism, the formalist claims that mathematics "works" in its own realm of existence. But the claim of "works" is sophistic deception, and the formalist really digs deeper into Platonism, hiding behind the smoke and mirrors of the sophistry hidden behind this word "works". That is when the term "mathemagician" is called for.

You finally said something interesting. Is the 5 in your mind the same as the 5 in my mind? I think so, but I might be hard pressed to rigorously argue the point. — fishfry

I believe, the concept of "five" in my mind is completely different from, though similar to, the concept of "five" in your mind. There is a number of ways to demonstrate the truth of this. The first is to get two different people to define the term, and see if they use the exact same expression. Another way is to look at what "five" means in different numbering systems, natural, rational, real, etc.. Another is from the discussions of mathematical principles in general. There is always difference in interpretation of such principles. You and I have significant differences, You and Tones have less significant differences.

Nevertheless, the differences exist and are very real. There is a principle which I've seen argued, and this is to say that this type of difference is a difference which does not make a difference. Aristotle called these differences accidentals, what is nonessential. The problem with that expression though, "difference which does not make a difference", is that to notice something as a difference, it is implied that it has already made a difference. So this argument is really nothing other than veiled contradiction.

Anyway, this is the issue with identity, in a nutshell. When we ignore differences which we designate as not making a difference, and say that two instances are "the same" on that basis, we really violate the meaning of "same". The meaning of "same" is violated because we know that we are noticing differences, yet dismissing them as not making a difference, in order to incorrectly say "same". Therefore we know ourselves to be dishonest with ourselves when we say that the two instances are the same, by ignoring differences which are judged as not making a difference. So when you say that you think the 5 in my mind is the same as the 5 in your mind, I think that this is an instance of dishonesty, you really know that there are differences, and if pressed to argue such a claim, you'd end up in contradiction, dismissing the obvious differences as not making a difference.

Is an apple an instance of fruit? Apples don't have a peelable yellow skin. 'Splain me this point. By this logic, nothing could ever be a specific instance of anything, since specific things always differ in some particulars from other things in the same class. — fishfry

Right, particulars are instances, specifics are not. The concept "red" is not an instance of colour, it is a specific type of colour. A particular red thing is an instance of red, and an instance of colour, exemplifying both. The concept "apple" is not an instance of fruit, it is a specific type of fruit. A particular apple is an instance of both. The concept "5" is not an instance of number, it is a specific type of number. A group of five particular things is an instance of both.

When I arrive home in the evening, it makes quite a big difference to me if I return to the same residence or just one that's "equal" to it in value. — fishfry

Hey fishfry, do you not remember what you said to me? You said " I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about." Now you've totally changed your position to say "it makes quite a big difference to me", if the taxi driver took you to a house which had an equal fare as yours, but was not the same house.

I have two different dollar bills in my pocket. They are not the same dollar bill. But they are equal in value.

(1)

bill 1 is not the same as bill 2
bill 1 does not equal bill 2
bill 1 is not identical with bill 2

Those are true and say the same thing as one another.

(2)

the value of bill 1 is the same as the value of bill 2
the value of bill 1 equals the value of bill 2
the value of bill 1 is identical with the value of bill 2

Those are true and say the same thing as one another.

(1) and (2) together is not a contradiction.

The valuation of a bill is a function, call it 'v'.

bill 1 does not equal bill 2
but
v(bill 1) equals v(bill 2)

Example from math:

Let v be the squaring function. So v(x) = x^2.

1 does not equal -1

but

v(1) = v(-1)

as

1^2 = -1^2

Mathematics adheres to the law of identity, since in mathematics, for any x, x=x, which is to say, for any x, x is x.

Further Adventures Of The Crank:

Sales Clerk: That will be five dollars.

/ The crank puts four one dollar bills on the counter /

Crank: There you go, five dollars.

Sales Clerk: That's only four dollars.

Crank: No, it's five dollars.

/ The sales clerk counts the bills by hand /

Sales Clerk: You see, only four dollars.

Crank: Five in your mind is different from five in my mind. I pay based only on what is in my mind. I gave you five dollars, now would you please put that copy of 'Hegel For Dummies' in a bag for me, as I just paid you five dollars for it?

Sales Clerk [on microphone]: Security at register seven. Security at register seven please.

Mathematics adheres to the law of identity, since in mathematics, for any x, x=x, which is to say, for any x, x is x. — TonesInDeepFreeze

Sorry Tones, but "for any x, x=x" does not say "for any x, x is x", unless "=" is defined as "is". And, in mathematics it is very clear that "=" is not defined as "is".

It is crystal clear that '=' is interpreted as 'is' in mathematics, since it is explicitly stated that '=' is interpreted as 'is' in mathematics. That the crank doesn't know anything about '=' in mathematics, or anything else about mathematics or logic doesn't entail that it is not the case that '=' is interpreted as 'is' in mathematics. Fortunately, what is the case about mathematics or logic is not affected by the crank's ignorance.

it is explicitly stated that '=' is interpreted as 'is' in mathematics. — TonesInDeepFreeze

Explicitly stated by you, the head sophist

= Equal sign ... equals ... Indicates two values
are the same -(-5) = 5
2z2 + 4z - 6 = 0

https://www.techtarget.com/searchdatacenter/definition/Mathematical-Symbols

'=' is interpreted:

For any terms 'T' and 'S'

T = S

is true

if and only if

the denotation of 'T' is the denotation of 'S'.

Consult any textbook in mathematical logic. The ignorant, confused, arbitrary personal dictates of the crank don't count.

Explicity stated in any textbook in mathematical logic.

Explicitly stated.... — Metaphysician Undercover

I'm jumping in here not because anyone needs my help, but instead because I have questions pending before you that you have not even attempted to answer, and because of your claims and lack of substantive response I hold you obliged to answer them. You will find them above on page 20 where you left them.

Mathematics adheres to the law of identity, since in mathematics, for any x, x=x, which is to say, for any x, x is x. — TonesInDeepFreeze

Are you referring to 'is' in terms of identity or value? For example: 5 is 5 in both mathematics and in our understanding of numerical systems. Meanwhile, £5 doesn't equal $5 or €5 because of the disparity in monetary value. Although every bill or note is represented by the payment of x5, it will depend on the value. So, x = x, doesn't equal to "is." To apply this, I need to carefully consider the specific context. Right? 

And, in mathematics it is very clear that "=" is not defined as "is". — Metaphysician Undercover

Could it be defined as "equals to..."?

(1) The symbols I used are common. The formulas I gave are not complicated. If one knows merely basic symbolic logical notation, then one can read right from my formulas into English. For example:

AxEy yex

reads as

For all x, there exists a y such that y is a member of x. — TonesInDeepFreeze

That is not my point. For whatever reason, I've always had difficulty with your posts. Maybe it's the symbols. Maybe it's the words. I don't know.

(2) I did give lots of explanations in certain contexts. — TonesInDeepFreeze

I'll concede that either I'm too dense to follow your arguments, or I just lack the logic background. I appreciate your efforts.

(3) You complain about the length of my posts, but also say I should give more explanation. You can't have it both ways. And you're hypocritical since your own posts are often long, and often enough have not merely a few symbols. — TonesInDeepFreeze

Guilty as charged on all counts.

I don't understand you. I gave you an example of how equivocation of "same" has a considerable effect. Of course it has no effect in "pure mathematics", because by definition "pure" mathematics maintains its purity, and the purity of its definitions. — Metaphysician Undercover

Then you DO understand me perfectly.

1) With regard to math, "same as", "=", and "is identical with" are synonymous. Period.

2) With respect to everything outside of math, you are probably right, but I take no position on it whatsoever. So if you're right about photons, that's great. Not any point I'm making or anything I care about in the context of this discussion.

Pure mathematics is not applied, and therefore has no effect in relation to the physical world where "same" means something else.. We live in the physical world, our cares and concerns involve the world we live in, it is impossible that anything in the fantasy world of "pure mathematics" could actually concern us. This is known as the interaction problem of idealism. However, in reality we apply mathematics and this is where the concerns are. — Metaphysician Undercover

This is of course true, but irrelevant to my part of this discussion. If you have some point to make about sameness as it pertains to playgrounds and photons, I'll concede the point without even thinking about it. Because I don't care about the issue. So if you're right, you're right. And even if you're wrong, I don't care, so I won't bother to try to prove you're wrong. Therefore I concede that you are right about a discussion I'm not even having with you.

You seem to misunderstand the issue completely. — Metaphysician Undercover

You agree with me about pure math. And I'm entirely agnostic about any other aspect. I can't misunderstand what I have no interest in understanding. I can be ignorant; but I can't be wrong, because I haven't even taken a position.

You appear to understand that there is a difference between the use of "same" by mathematicians (synonymous with equal), and the use of "same" in the law of identity (not synonymous with equal). — Metaphysician Undercover

Not as it's understood in math. You're just trying to argue with me about something that I have no opinion on and no interest in having an opinion. I know, for an opinionated guy like me that's unusual. But there are a lot of things I have no opinion on.

You said that this difference has no bearing on anything you know or care about. The things included in the category of what you know and care about, are not limited to principles of pure mathematics, because you live and act in the physical world. The law of identity applies to things in the physical world which we live and act in. — Metaphysician Undercover

I don't get worked up about this at the grocery store. For example one onion is pretty much the "same" as any other, and the ways in which they are the same and the ways in which they are different may be of interest to a philosopher. But I don't trouble myself with it. I just pick out some onions.

So, to make myself clear — Metaphysician Undercover

If only.

I do not claim that there is a problem with using "same" as synonymous with equal, within the conceptual structures of mathematics. — Metaphysician Undercover

Well then we are DONE. You have agreed with my point, with my ONLY point. I have no other point. I have no other opinion. I have no other thesis. I concede everything you say about the subject, not because you're necessarily right, but because I just don't care one way or the other.

So we're done. You agree with me about the only point I'm making. Which I appreciate a lot, actually. It's taken me a long time to get you to agree that "same" and "=" are synonymous in math.

The problem is in the application of mathematics, as inevitably it is applied, and this use of "same" is brought into the world of physical activity, and taken to be consistent with the use of "same" when referring to physical objects. — Metaphysician Undercover

I care not about applications. Nor do I believe that you are correct on this point. Math is math, whether you apply it or not.

That is where the problem occurs. Sophjists such as Tones enhance and deepen the problem by arguing that the use of "same" in mathematics(synonymous with equal) is consistent with the use of "same" in the law of identity (not synonymous with equal). — Metaphysician Undercover

I can't comment on the thoughts of others. Nor do I understand much of what @TonesInDeepFreeze says. But he did clarify a point of longstanding confusion on my part about the axiom of extensionality, for which I'm grateful to him, even if he thinks I'm still misunderstanding.

This is exactly the issue, the reality of the situation is that we do have it both ways. There are two very distinct ways for understanding "same". You can dictate "you cannot have it both ways" all you want, but that's not consistent with reality, where we have both ways of using the term. If you think that we ought to reduce this to one, (insisting "we cannot have it both ways"), the two cannot be combined, or reduced to one, because they are fundamentally incompatible (despite what the head sophist claims). This means that we have to choose one or the other. If we choose the one from pure mathematics, then we have nothing left to understand the identity of a physical object in its temporal extension. If we choose the one from the law of identity, then we simply understand "equal" as distinct from "same", and the problem is solved. Obviously the latter makes the most sense, and doing this would support your imperative dictate: "You can not have it both ways." — Metaphysician Undercover

You have conceded my point regarding math. I have no other point. I don't worry about sameness for onions or photons. I can chop up an onion like nobody's business; and I can flip on a light switch. That's as far as I go with onions and photons.

It is very clearly you are the one who does not understand how math "works". — Metaphysician Undercover

Sadly I came to this conclusion in grad school.

Math only works when it is applied. — Metaphysician Undercover

I don't know why you make demonstrably absurd claims like that.

"Pure mathematics" does nothing, it does not "work", as math only works in application. — Metaphysician Undercover

Tens of thousands of professional pure mathematicians would disagree.

You are only fooling yourself, with this idea that pure mathematics is completely removed from the physical world, the world of content, and it "works" within its own formal structures. — Metaphysician Undercover

Yeah yeah.

That is the folly of formalism which I explained earlier. — Metaphysician Undercover

I'm only a formalist sometimes, when it helps clarify an argument. Clearly math is "about" something. Just not clear exactly what.

To avoid the interaction problem of Platonist idealism, the formalist claims that mathematics "works" in its own realm of existence. But the claim of "works" is sophistic deception, and the formalist really digs deeper into Platonism, hiding behind the smoke and mirrors of the sophistry hidden behind this word "works". That is when the term "mathemagician" is called for. — Metaphysician Undercover

I agree with you that most mathematicians are Platonists. A group theorist does not believe he's merely pushing symbols. He's discovering facts about groups. I concede your point. But I'm not concerned with it.

I believe, the concept of "five" in my mind is completely different from, though similar to, the concept of "five" in your mind. There is a number of ways to demonstrate the truth of this. The first is to get two different people to define the term, and see if they use the exact same expression. — Metaphysician Undercover

Any two set theorists will give {0, 1, 2, 3, 4} as the definition of 5. That's due to John von Neumann, who invented game theory, worked on quantum physics, worked on the theory of the hydrogen bomb, and did fundamental work in set theory. Now there was a guy who blended the applied with the pure.

Another way is to look at what "five" means in different numbering systems, natural, rational, real, etc.. Another is from the discussions of mathematical principles in general. There is always difference in interpretation of such principles. You and I have significant differences, You and Tones have less significant differences.[/quotet]

You raise an interesting point. The integer 5 and the real number 5 are completely different sets. They are NOT the same set at all. They are not equal as sets. But they are the "same" number, for the reason that we can embed the integers inside the reals in a structure-preserving manner. This raises issues of structuralism in mathematics. Lot of interesting issues. Point being is that sameness as sets is NOT actually the basis of sameness in mathematics, entirely contrary to what I've been claiming. There are structural or categorical ways of looking at sameness. I concede your point.

I'll give an example. The set of numbers {0, 1, 2, 3}, along with addition mod 4, is a cyclic group with four elements. Addition mod 4 just means that we only consider remainders after division by 4, so that 2 + 3 = 1. Hope that's clear.

Now you may know the imaginary unit $i$, characterized by the property that $i[&2 = -1/math]. Then <span class="plush-jax" style="font-size:15px"><script type="math/tex">i^3 = -i$, and $i^4 = 1$. So the set of complex numbers $\{i, -1, -i, 1\}$, with the operation of complex number multiplication, is also a cyclic group of order 4. But as any group theorist will tell you, there is only one cyclic group of order 4. Or to put it another way, any two cyclic groups of order 4 are isomorphic to each other.

So these two groups, the integers mod 4 and the integer powers of $i$, are the exact same group, even though they are ridiculously different as sets.

This is a pretty good introduction to structuralism in math. What mathematicians are studying is not the particular sets; but rather, the abstract structure of which these two sets are each representatives. What group theorists care about is the idea of a cycle of four things. How we represent the cycle doesn't matter. Now that's Platonism too, because the cyclic group of order 4 is "out there" in the abstract world of patterns. It's real. It's not a set, it's merely represented in various ways by sets.

So you are right that sameness is a tricky business, even in math. Perhaps I will need to retreat to saying that sameness and set equality are synonymous for sets. For groups, that's not true. Different sets can represent the same group.

Maybe I just talked myself into your point.
— Metaphysician Undercover

Nevertheless, the differences exist and are very real. There is a principle which I've seen argued, and this is to say that this type of difference is a difference which does not make a difference. — Metaphysician Undercover

I like that.

Aristotle called these differences accidentals, what is nonessential. The problem with that expression though, "difference which does not make a difference", is that to notice something as a difference, it is implied that it has already made a difference. So this argument is really nothing other than veiled contradiction. — Metaphysician Undercover

Can you give an example? I might have not followed you.

Anyway, this is the issue with identity, in a nutshell. When we ignore differences which we designate as not making a difference, and say that two instances are "the same" on that basis, we really violate the meaning of "same". The meaning of "same" is violated because we know that we are noticing differences, yet dismissing them as not making a difference, in order to incorrectly say "same". Therefore we know ourselves to be dishonest with ourselves when we say that the two instances are the same, by ignoring differences which are judged as not making a difference. — Metaphysician Undercover

Like the two ways I showed of representing the cyclic group of four elements? So are you talking about structuralism? Two things can be the "same" if they have the same structure, even if they are very different as things.

So when you say that you think the 5 in my mind is the same as the 5 in your mind, I think that this is an instance of dishonesty, you really know that there are differences, and if pressed to argue such a claim, you'd end up in contradiction, dismissing the obvious differences as not making a difference. — Metaphysician Undercover

Well ... no. There is only one Platonic 5. We may all think of it different ways, but we all have the same concept of fiveness. So yes and no to your point.

Right, particulars are instances, specifics are not. — Metaphysician Undercover

Particulars versus instances. I take your point there too. I think. Fruit and apples, colors and red. Two subtly different concepts. Which is which?

The concept "red" is not an instance of colour, it is a specific type of colour. — Metaphysician Undercover

Ok. I accept that.

A particular red thing is an instance of red, and an instance of colour, exemplifying both. — Metaphysician Undercover

\

Hmmm. Ok.

The concept "apple" is not an instance of fruit, it is a specific type of fruit.[/quote[

Ok
— Metaphysician Undercover

A particular apple is an instance of both. — Metaphysician Undercover

Ok. Like apple is a subclass of fruit, but a specific apple in my hand is an instance of the class of apples. Is that right?

The concept "5" is not an instance of number, it is a specific type of number. — Metaphysician Undercover

A type of number. No, don't agree. Real numbers and complex numbers and quaternions are types of numbers. The real number 5 is an instance of a real number hence an instance of a number. It must be so, mustn't it?

A group of five particular things is an instance of both. — Metaphysician Undercover

I do not agree that five apples is an instance of the number 5. The collection of five apples has cardinality 5. That is, 5 is an attribute of the group of five apples. Numbers are pure, they are not apples. A group of apples is not an instance of 5. It might be a representation of the number 5, I could live with that, in the sense that xxxxx is a representation of the number 5.

Hey fishfry, do you not remember what you said to me? You said " I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about." Now you've totally changed your position to say "it makes quite a big difference to me", if the taxi driver took you to a house which had an equal fare as yours, but was not the same house. — Metaphysician Undercover

I already addressed this point.

I said in the context of doing math blah blah. When I'm not doing math, the subtleties of sameness matter. Of course the subtleties matter in math too, as I just showed. Two different sets can represent the same group. These days structuralism is how we think about things in math. Two things are the same when they have all the same relationships to all the other things.

Maybe it's the symbols. Maybe it's the words. — fishfry

The symbols are standard. The words are ordinary for logic and mathematics, or if personal, they're defined.

So maybe it's something else.

Most glaringly of all, what accounts for you recently claiming that I hadn't specified 'identity theory' when I had specified it multiple times in this thread, including multiple times addressed to you, and even twice quoted by you? Your claim is bizarre.

Are you referring to 'is' in terms of identity or value? — javi2541997

Identity. I don't know how to make it more clear than I already have.

x is x.

With models:

'=' is interpreted as {<x x> | x e U} where U is the universe for the model

Although every bill or note is represented by the payment of x5, it will depend on the value. So, x = x, doesn't equal to "is." — javi2541997

5 = 5

5 dollars not= 5 pounds

If you have 5 dollars and I have 5 pounds, then the number of your dollars is the number of my pounds. 5 is 5. But that does not entail that 5 pounds is 5 dollars.

The number of things is not the things.

'x = x' doesn't equal 'is'. Rather, '=' stands for 'is', so 'x = x' stands for 'x is x'.

None of that is a contradiction or problem.

Identity. — TonesInDeepFreeze

The number of things is not the things. — TonesInDeepFreeze

:up:

Explicity stated in any textbook in mathematical logic. — TonesInDeepFreeze

This is exactly the problem. Notice you refer to "any textbook in mathematical logic", rather than any textbook in mathematics. If you look at a textbook in mathematics, you'll find "=" defined in the way of my reference, "equals"... "indicates two values are the same". So there is a discrepancy between what "=" means in mathematics, and what "=" means in mathematical logic. And, since mathematical logic is supposed to provide a representation of the logic used in mathematics, we can conclude that mathematical logic has a false premise. The proposition of mathematical logic, which states that "=" indicates identity, or that it means "is" or "is the same as" in mathematics, is a false premise. This is a false representation of how "=" is used in mathematics. As I explained in other threads, if "=" indicated "is the same as", then an equation would be absolutely useless, because the left side would say the exact same thing as the right side. (Incidentally, this is what many philosophers have been known to say about the law of identity, it is a useless tautology). In ontology though we see the law of identity as a useful tool against sophistry like yours.

You agree with me about pure math. — fishfry

I agreed with you about "pure math", for the sake of discussion, so that we could obtain some understanding of each other. But I will tell you now, as came up one other time when we had this discussion, I do not agree that there is such a thing as "pure math" by your understanding of this term. So I agree that if there was such a thing as pure math, that's what it would be like. However, I think your idea of "pure math" is just a Platonist/formalist fantasy, which is a misrepresentation of what mathematics is. In reality, all math is corrupted by pragmatics to some degree, and none reaches the goal of "pure math". You criticize me to say, it's not a goal, it's what pure math is, but I say that's false, it is a goal, an ideal, which cannot be obtained. Therefore "pure math" as you understand it, is not real, it's an ideal.

I think the issue being exposed here is a difference of opinion as to what mathematics is. Since this is a question of "what something is", the type of existence it has, I think it is an ontological issue. Would you agree with this assessment? For example, the head sophist refers to "mathematical logic", and I find this defined in Wikipedia as the study of the formal logic within mathematics. So we have a distinction here between the use of mathematics (applied mathematics), and the study of the logic used by mathematicians (mathematical logic). "Mathematical logic" would be a sort of representation, or description, of the logic used in mathematics. What you call "pure mathematics", I believe would be something distinct from both, applied math and mathematical logic, as the creative process whereby mathematical principles are developed. But I think that this process is not really "pure", it's always tainted by pragmatics and therefore empirical principles.

The issue I have with the head sophist is with the way that mathematical logic represents the use of the = symbol as an identity symbol. In applied mathematics, it is impossible that "=" is an identity symbol because if both sides of an equation represented the exact same thing, the equation would be absolutely useless. This I've explained in a number of different threads. In reality, as any mathematics textbook will show, "=" means "has the same value as". Therefore we can conclude that any mathematical logic which represents "=" as an identity symbol is simply using a false proposition. When a "textbook in mathematical logic" states that "=" is an identity symbol, this can be taken as the false premise of mathematical logic.

You have conceded my point regarding math. I have no other point. — fishfry

I have conceded the point regarding what you call "pure math". However, I am now qualifying this concession to say that "pure math" is just an unreal ideal. There is no such thing as pure math. It's a term which people like to use in an attempt to validate their ideals. In reality though, such ideals are fiction, so all that I have really conceded, is that within the fictitious conception which you call "pure math", this is the way things are. Of course, I'm not going to argue about the way things are in your work of fiction, but I will argue about the way that your fiction bears on the real world.

Tens of thousands of professional pure mathematicians would disagree. — fishfry

Sure, there are thousands of people who might call themselves "pure mathematicians". In reality though, these people are not engaged in "pure mathematics", as I believe you understand this to mean. As I said above, all mathematics is tainted by pragmatics (applications), and there is no such thing as "pure" mathematics.

This is very evident in our discussion of the meaning of "=". In what you call "pure mathematics", we might say that "=" signifies "is the same as". This would remove the basic fact that what mathematicians work with are values. To make the mathematics "pure" we must remove this content, what the mathematicians work with, values. We remove the inherent nature of the thing represented by the symbols (i.e. that the symbols represent values) to allow simply that the symbols represent things without any inherent nature, no inherent content. Then we might claim the left side of the equation represents the exact same thing as the right. However, this type of equation would be totally useless. We could do nothing with an equation, solve no problems.

Furthermore, there would be a disconnect, an inconsistency between the mathematicians practising "applied" math, who use "=" to represent "is the same value as", and those "pure" mathematicians creating mathematical principles which were inconsistent with the applied mathematics. Since the supposed "pure mathematicians" actually produce principles which are compatible with, and are actually used in applied mathematics, we can conclude that the supposed "pure" mathematics is not really pure, and the principles they are using and developing do not really treat "=" as meaning "is the same as". That's just a misrepresentation, supported by the misrepresentation that these people are doing "pure" mathematics.

Any two set theorists will give {0, 1, 2, 3, 4} as the definition of 5. That's due to John von Neumann, who invented game theory, worked on quantum physics, worked on the theory of the hydrogen bomb, and did fundamental work in set theory. Now there was a guy who blended the applied with the pure. — fishfry

I can't say I understand everything you wrote following this, but it mostly makes sense to me. I'll have to work on these ideas of "mod 4", and "cyclical group".

Can you give an example? I might have not followed you. — fishfry

What I mean, is that if you recognize that two things are different from each other, then that difference has already made a difference to you (in the subconscious for example) by the very fact that you are recognizing them as different. So for example, if you see two chairs across the room, and they appear to be identical, yet you see them as distinct, then the difference between them must have already made a difference to you, by the fact that you see them as distinct. So to say that the difference between them is a difference which doesn't make a difference must be a falsity from the outset. We might even say that they are identical in every way except that they are in different locations, but this very difference is the difference which makes them two distinct chairs instead of one and the same chair.

A type of number. No, don't agree. Real numbers and complex numbers and quaternions are types of numbers. The real number 5 is an instance of a real number hence an instance of a number. It must be so, mustn't it? — fishfry

I knew you wouldn't agree, but i wouldn't agree that the real number 5 is an instance of a real number. The problem I think has to do with the statement "a real number". "The real numbers" is a conceptual construct in itself. This conception dictates the the meaning of "a real number". So in reality any supposed instance of "a real number" is just a logical conclusion drawn from the dictates of "the real numbers". In other words its not a distinct or individuated thing, which would be required for "an instance", it's just a specific part of "the real numbers". Can we agree that the real number 5 is a specific real number?

Mathematical logic formalizes the logic used in other mathematics. The explication of '=' in mathematical logic conforms to the use in mathematics.

The crank says that we may look in a textbook in mathematics to see that the definition of '=' differs from mathematical logic. What specific textbook does the crank refer to?

"two values are the same"

Indeed, in both mathematical logic and in other mathematics:

'1+1 = 2' means that the value of the expression '1+1' is the same as the value of the expression '2'.

'=' is interpreted:

For any terms 'T' and 'S'

T = S

is true

if and only if

the denotation of 'T' is the denotation of 'S'. — TonesInDeepFreeze

The crank claims that we may look in a textbook in mathematics to see that mathematics doesn't agree. What textbooks are those?

And notice that the crank has shifted his argument. Previously he recognized that mathematics regards '=' as 'is the same as' and that mathematics is wrong to do that, but now he's claiming mathematics doesn't but that it is only mathematical logic that does, modulo his latest tact of blaming only pure mathematics. The crank is always greased for easy shifting.

Most simply, when we say "1+1 = 2" we mean that '1+1' and '2' name the same number. Or we are to believe they name different numbers? What numbers are those?

Clerk: Okay, we have one plus one plus one plus one. Okay, that is four. At $2 each, that's $8.

The Crank: No, one plus one plus one plus one is not four. I only pay for one plus one plus one plus one, not for four

Clerk [into mic]: We have a problem at register ten.

So maybe it's something else. — TonesInDeepFreeze

My deficiency, I'm sure.

Most glaringly of all, what accounts for you recently claiming that I hadn't specified 'identity theory' when I had specified it multiple times in this thread, including multiple times addressed to you, and even twice quoted by you? Your claim is bizarre. — TonesInDeepFreeze

I'll retract it then, as an alternative to arguing the point. Or if you consider the second clause as adding fuel to the fire, I'll retract it.

I agreed with you about "pure math", for the sake of discussion, so that we could obtain some understanding of each other. But I will tell you now, as came up one other time when we had this discussion, I do not agree that there is such a thing as "pure math" by your understanding of this term. So I agree that if there was such a thing as pure math, that's what it would be like. However, I think your idea of "pure math" is just a Platonist/formalist fantasy, which is a misrepresentation of what mathematics is. In reality, all math is corrupted by pragmatics to some degree, and none reaches the goal of "pure math". You criticize me to say, it's not a goal, it's what pure math is, but I say that's false, it is a goal, an ideal, which cannot be obtained. Therefore "pure math" as you understand it, is not real, it's an ideal. — Metaphysician Undercover

Those corrupt math professors. Something must be done. Pure math is math done without any eye towards contemporary applications. That's a decent enough working definition. Mathematicians know the difference.

I think the issue being exposed here is a difference of opinion as to what mathematics is. — Metaphysician Undercover

Not really. Nailing down a definition is unimportant. Mathematics is whatever mathematicians do in their professional capacity. Circular, but as good a definition as any. What difference does the definition of mathematics make?

Since this is a question of "what something is", the type of existence it has, I think it is an ontological issue. Would you agree with this assessment? — Metaphysician Undercover

No. Mathematics is a historically contingent human activity. It's different every day, every time someone publishes a new paper.

For example, the head sophist refers to "mathematical logic", and I find this defined in Wikipedia as the study of the formal logic within mathematics. So we have a distinction here between the use of mathematics (applied mathematics), and the study of the logic used by mathematicians (mathematical logic). "Mathematical logic" would be a sort of representation, or description, of the logic used in mathematics. What you call "pure mathematics", I believe would be something distinct from both, applied math and mathematical logic, as the creative process whereby mathematical principles are developed. But I think that this process is not really "pure", it's always tainted by pragmatics and therefore empirical principles. — Metaphysician Undercover

You must think I'm sticking to some pure/applied distinction. I'm not.

The issue I have with the head sophist — Metaphysician Undercover

Repeatedly calling a fellow forum member that makes you look like an asshole.

is with the way that mathematical logic represents the use of the = symbol as an identity symbol. In applied mathematics, it is impossible that "=" is an identity symbol because if both sides of an equation represented the exact same thing, the equation would be absolutely useless. — Metaphysician Undercover

This is a standard complaint. If math follows from axioms, then all the theorems are tautologies hence no new information is added once we write down the theorems. But that's like saying the sculptor should save himself the trouble and just leave the statue in the block of clay. Or that once elements exist, chemists are doing trivial work in combining them. It's a specious and disingenuous argument.

This I've explained in a number of different threads. — Metaphysician Undercover

Repeating a mistake is no virtue.

In reality, as any mathematics textbook will show, "=" means "has the same value as". — Metaphysician Undercover

We agreed long ago that 1 + 1 and 2 are not the same string; and many people have explained the difference between the intensional and extensional meanings of a string. Morning star and evening star and all that.

Therefore we can conclude that any mathematical logic which represents "=" as an identity symbol is simply using a false proposition. When a "textbook in mathematical logic" states that "=" is an identity symbol, this can be taken as the false premise of mathematical logic. — Metaphysician Undercover

What math teacher hurt your feelings, man? Was it Mrs. Screechy in third grade? I had Mrs. Screechy for trig, and she all but wrecked me. It's over half a century later and I can still hear her screechy voice. I hated that woman, still do. When I'm in charge, I'm sending all the math teachers to Gitmo first thing.

I have conceded the point regarding what you call "pure math". However, I am now qualifying this concession to say that "pure math" is just an unreal ideal. — Metaphysician Undercover

You are the one making a big deal out of the distinction. Besides, in my last post I ended up talking myself out of my entire thesis due to the structuralist turn in mathematics.

There is no such thing as pure math. It's a term which people like to use in an attempt to validate their ideals. In reality though, such ideals are fiction, so all that I have really conceded, is that within the fictitious conception which you call "pure math", this is the way things are. Of course, I'm not going to argue about the way things are in your work of fiction, but I will argue about the way that your fiction bears on the real world. — Metaphysician Undercover

I really haven't made a big deal of the distinction; and if I did, I shouldn't have.

Sure, there are thousands of people who might call themselves "pure mathematicians". In reality though, these people are not engaged in "pure mathematics", as I believe you understand this to mean. As I said above, all mathematics is tainted by pragmatics (applications), and there is no such thing as "pure" mathematics. — Metaphysician Undercover

A piece of math without any known application is pure math. A hundred years from now it could be applied. The most striking case is number theory, which was totally useless for 2000 years then became the basis of public key cryptography in the 1980s, the basis of online commerce.

This is very evident in our discussion of the meaning of "=". In what you call "pure mathematics", we might say that "=" signifies "is the same as". This would remove the basic fact that what mathematicians work with are values. To make the mathematics "pure" we must remove this content, what the mathematicians work with, values. We remove the inherent nature of the thing represented by the symbols (i.e. that the symbols represent values) to allow simply that the symbols represent things without any inherent nature, no inherent content. Then we might claim the left side of the equation represents the exact same thing as the right. However, this type of equation would be totally useless. We could do nothing with an equation, solve no problems. — Metaphysician Undercover

I just think you're working yourself up over nothing. I'm losing interest. Can you write less? This is tedious, I find nothing of interest here.

Furthermore, there would be a disconnect, an inconsistency between the mathematicians practising "applied" math, who use "=" to represent "is the same value as", and those "pure" mathematicians creating mathematical principles which were inconsistent with the applied mathematics. Since the supposed "pure mathematicians" actually produce principles which are compatible with, and are actually used in applied mathematics, we can conclude that the supposed "pure" mathematics is not really pure, and the principles they are using and developing do not really treat "=" as meaning "is the same as". That's just a misrepresentation, supported by the misrepresentation that these people are doing "pure" mathematics. — Metaphysician Undercover

Whatevs. I can't follow you. And I've already noted that the difference between pure and applied math is often a century or two, or a millennium or two.

I can't say I understand everything you wrote following this, but it mostly makes sense to me. I'll have to work on these ideas of "mod 4", and "cyclical group". — Metaphysician Undercover

Sorry. Nevermind all that. Point being that two groups are essentially the same even though their presentation as sets is entirely different. Just like the real number 5 and the integer 5 are essentially the same even though they are different sets. Now what do I mean by "essentially the same?" Well now we're into structuralism and category theory. Sameness in math is a deep subject. I'll take your point on that.

What I mean, is that if you recognize that two things are different from each other, then that difference has already made a difference to you (in the subconscious for example) by the very fact that you are recognizing them as different. So for example, if you see two chairs across the room, and they appear to be identical, yet you see them as distinct, then the difference between them must have already made a difference to you, by the fact that you see them as distinct. So to say that the difference between them is a difference which doesn't make a difference must be a falsity from the outset. We might even say that they are identical in every way except that they are in different locations, but this very difference is the difference which makes them two distinct chairs instead of one and the same chair. — Metaphysician Undercover

Ok. They're two instances of chair-ness.

I knew you wouldn't agree, but i wouldn't agree that the real number 5 is an instance of a real number. — Metaphysician Undercover

I can't even imagine how it isn't.

The problem I think has to do with the statement "a real number". "The real numbers" is a conceptual construct in itself. — Metaphysician Undercover

Yes.

This conception dictates the the meaning of "a real number". So in reality any supposed instance of "a real number" is just a logical conclusion drawn from the dictates of "the real numbers". — Metaphysician Undercover

Even so, 5 is one of the real numbers. What do you call it if not an instance? What WOULD be an instance of a real number?

In other words its not a distinct or individuated thing, which would be required for "an instance", it's just a specific part of "the real numbers". Can we agree that the real number 5 is a specific real number? — Metaphysician Undercover

I agree that 5 is a specific real number. But ok, instance means something else. Still a bit fuzzy but sort of seeing your point.

'1+1 = 2' means that the value of the expression '1+1' is the same as the value of the expression '2'. — TonesInDeepFreeze

Why does it take so long to understand an axiom that appears so simple?

I once read Bertrand Russell's works, and one of them was Principia Mathematica. Well, it took him and another colleague of his more than 300 pages to prove that 1+1 = 2. I acknowledge that I struggled to understand some of their pages and axioms due to my lack of familiarity with logical language. There was even some criticism of the work of Russell and Whitehead because it seems the work was based on finding 'truth logic' 

I asked myself then. Is 1 + 1 = 2 a logical truth? And I found on the Internet big debates among mathematicians and logicians about whether it is a tautology, a logical truth, or a theorem.

In the following link (1 + 1 = 2) you will see similar answers to yours: 1 + 1 = 2 is a 'definition'.
2 is another way of defining '1 + 1' if I am not mistaken...

Is 1 + 1 = 2 a logical truth? — javi2541997

Frege proposed a way that it would be a logical truth. But his way was inconsistent.

1 + 1 = 2 is a 'definition'.
2 — javi2541997

That's often the case, and per the definition, '1+1 = 2' means that '1+1' names the name number as is named by '2'. 1+1 is 2.

I'll retract it then, as an alternative to arguing the point. Or if you consider the second clause as adding fuel to the fire, I'll retract it. — fishfry

It is bizarre to suggest there's any arguing the point, when the point has been so profusely documented. Your retraction and your offer to retract the bizarre qualifier in the retraction are a self-serving and sneaky way to put the ball back in my court where it doesn't belong.

Frege proposed a way that it would be a logical truth. But his way was inconsistent. — TonesInDeepFreeze

Isn't that a bit too much to put on the Basic Law V?
If we have problems with infinite sets, why would you throw away also everything finite?

How about Peano axioms or Peano Arithmetic?
Are they inconsistent also according to you?