The best introduction to the subject I have found is in the introduction to Church's 'Introduction To Mathematical Logic', as indeed that whole introduction is a quintessential primer for the basics of logic.

It's not required that each concept, each abstraction itself corresponds to a particular concrete. — TonesInDeepFreeze

I appreciate your post.

Yes, the mind has a framework of concepts, such as beauty, infinity, pain, mountain, house, government, addition, multiplication, sky, salt, which we use to organise our concrete experiences.

On the one hand, for example, our concept of mountain refers to an abstract object, in that it does not refer to one particular mountain, but mountains in general. But on the other hand, there must be an intentionality to our concepts, in that the mind is not able to comprehend a concept without thinking about something concrete, whether a particular object, such as the Mont Blanc, a particular process, such as adding more height to a high hill or the particular word itself, "mountain".

That is not to say that each concept can only have one concrete instantiation, but rather each time I think of a mountain my concrete instantiation may be different, and different again for anyone else who thinks of a mountain.

My point is that I agree that it is not the case that an abstract concept corresponds to one particular concrete instantiation, but rather we can only understand an abstract concept by thinking of some concrete instantiation of it, which may be a concrete object (Mont Blanc), a concrete process (addition) or a concrete word ("mountain").

That sounds much like my thinking on days that do not start with the letter "T". Abstractions as a fabrication of grammar...

Increasingly I find learning novel logic systems quite difficult, as if there is too much background missing. The topic is surprisingly different to when I first studied it, much more of an emphasis on computing, far more integrated than it once was, and my short term memory is not what it was fifty years since. When in a masochistic frame of mind I'll work through bits of the Open Logic text. It's often very simple things that hold one up - it seems to be the text of choice for undergrad logic courses and for some of the more advanced stuff. There is a lot to be said for the discipline of having a tutor to help work through examples, especially where small bits of jargon catch one out.

Church is an interesting choice.

One may reject ideation and communication premised in abstract objects. But the notion of identity is not even limited to abstract objects. Whatever things one does countenance as existing, named by, say, T and S, we have T = S if and only if T is S. That is what '=' means when it is used in contexts of ordinary identity theory, logic, mathematics and other contexts to. If one wishes to use it with another meaning in another context, then, of course, fine. But that doesn't justify saying that in logic and mathematics it is not used just as logic and mathematics says it is used. — TonesInDeepFreeze

The sense of "identity" I am concerned with is that stated by the law of identity, "a thing is the same as itself". Do you agree with this formulation of the law of identity, and that if logic and mathematics uses "identity" in a way which is inconsistent with this, then logic and mathematics violate the law of identity?

Again, more exactly:

If 'T' and 'S' are terms, then

'T = S' is true if and only if T is S. — TonesInDeepFreeze

I would accept this as consistent with the law of identity, if we're careful to clarify that what we are talking about is the thing which "T" and "S" each signify. Clearly T itself, as a symbol, is not the same as S as a symbol.

And whether 'T' and 'S' stand for abstract things, abstract objects, values that are abstract things, values that are abstract objects, concrete things, physical things, or whatever things you are looking at right now on your desk. — TonesInDeepFreeze

The problem with this statement, is that a careful analysis and thorough understanding of what is here called "abstract things" will reveal that abstractions cannot be adequately understood as things with identity. So all these so-called "internal objects", conceptions, ideas, values, emotions, feelings, and everything else in this category, cannot be assumed to have an identity. This issue is extensively reviewed by Wittgenstein in The Philosophical Investigations. Particularly relevant is the part commonly known as the private language argument, where Wittgenstein provides the example of an attempt to assign the symbol "S" to a sensation. What is revealed is that "the sensation" cannot be known as having an identity. And this principle is extended by Wittgenstein to include all supposed "internal objects".

Due to what has been revealed by a large body of philosophical work in the past, I propose to you that if "T" and "S" are intended to stand for abstractions, conceptions, or anything else in this category commonly known as "internal", then "T" and "S" have no proper identity, as demonstrated by Wittgenstein. This was extensively covered by Aristotle under the concept of "substance", when he noticed the need to apply the law of identity against the sophistical arguments of Pythagorean idealists. Allowing that abstractions are identifiable things breaks down the categorical separation between ideas and things, allowing that the universe is composed of ideas.

Then, '1+1' refers the SUM of the number one with the number one. — TonesInDeepFreeze

This is incorrect, and this incorrectness I already explained to Michael. It is very clear that "1+1" refers to a specific operation which is indicated by "+". If we ignore this, and take a shortcut, assuming that the operation has already been carried out, and assume that "1+1" refers to the sum, then we ignore the role of "correctness" in the carrying out of the operation. Then one could stipulate any arbitrary expressions as referring to the same thing. I could say "1+1 = 8-2", and have my own private operations which produce this identity. In reality it is only through the means of carrying out the correct operation which is specifically signified by "+", that "1+1" can be said to be equal to "2". Therefore the meaning of "+" in that expression "1+1" is extremely significant to the meaning of the expression. It is intensional, and this intensionality cannot simply be taken for granted in the interpretation, to claim that the expression is extensional.

'1+1' does not stand for an operation. It stands for the result of an operation applied to an argument. — TonesInDeepFreeze

Obvious falsity. We read "1+1" as it is written, we don't read the implied result, "2". If what you said is true, then there would be no place for the learning of mathematics. We would not be able to account for the person who can read a mathematical expression, but cannot properly apply the principles required to produce the correct answers.

The truth of the matter is that the ability to correctly produce the answer, from the expressed operation, must be accounted for. It is simply not the case that a person goes from reading "1+1" as one plus one, to reading it as two, without a learning process, and that means acquiring the intensionality. The reality of this learning process, and how to properly account for it, is what Plato looked at in his theory of recollection, and what Wittgenstein looked at in The Philosophical Investigations.

And, I request that you please be honest with yourself. Do you really believe that you read the left side of an equation as the result of the expressed operation? That's simply not true, it's impossible because some operations are not carried out in the order that they appear. That is why I request honesty from you, and recognition that what is expressed by "1+1" is an operation to be carried out, not the SUM of that operation.

It is difficult to reason with someone about mathematics who doesn't understand that 1+1 is 2. — TonesInDeepFreeze

That goes two ways. When a person such as yourself, unwaveringly insists that the right side of an equation signifies the very same thing as the left side, despite a world full of applied mathematics as evidence to the contrary, then it becomes very difficult to reason with this person. The person simply refuses to look at all the evidence, and denies the evidential status of the evidence. The simple fact is that all the mathematical evidence supports what I say, so I am justified in my stance. But there is nothing but a stipulated "axiom" which supports your stance.

@Banno@MichaelThe issue we've encountered is that the axiom of extensionality is simply false. Of course, some will say that truth and falsity are not applicable judgements for mathematical axioms, and that is exactly why the axiom of extensionality is an ontological principle rather than a mathematical axiom.

What this so-called axiom attempts to do is to introduce truth and falsity into mathematics in the form of correspondence. It implies that there is an identified object which corresponds with the expressions of "1+1" and "3-1", replacing the true representation of 'correct answer' with this proposed corresponding "object". Now we'd have an "object" which corresponds with "1+1", as a form of truth, just like there is an object which correspond with "Mark Twain", as a form of truth.

This is why the axiom of extensionality is not a mathematical axiom, it is an ontological principle. Therefore it ought to be judged in a way which is appropriate to ontology.

The crank will mangle what I wrote, misrepresent it, presume to knock down strawmen of it. Likely, I won't have to time to compose a response, especially to the sheer volume of his confusions. — TonesInDeepFreeze

I don't see any strawmen, you just demonstrate a simple misunderstanding of how "=" is used in mathematics, and an equally simple refusal to seriously consider the evidence, resulting in a simple denial. Perhaps it would help you if we move on to more complex equations. Do you really believe that "2πr" signifies the very same concept as "the circumference of a circle"? Surely you recognize that "r" signifies a straight line, and "circumference" signifies a curved line, and by no stretch of the mathematical imagination do these two expressions represent the exact same thing. A curved line cannot be made to be compatible with a straight line, as indicated by the fact that pi is irrational.

some will say that truth and falsity are not applicable judgements for mathematical axioms — Metaphysician Undercover

Which is correct.

What this so-called axiom attempts to do is to introduce truth and falsity into mathematics in the form of correspondence. — Metaphysician Undercover

Maybe for mathematical realism, but then that’s a problem with mathematical realism. Just be a mathematical antirealist and accept that “true” in the context of maths just means something like “follows from the axioms”, with the axioms themselves not being truth-apt.

Referring back to this, it makes no sense to say that the axiom is either true or false. It just is an axiom, and the inference follows.

You’re making a mountain out of nothing.

The problem is that both you and Corvus badly misrepresent Wittgenstein in an attempt to subjugate his name to your psycoceramics.

So far neither of you have been able to cite anything like an endorsement of either your eccentric and unsound view of equity nor Corvus' confusing finite and infinite. Nor will you. — Banno

Under your thinking, anyone not thinking the same as you is misattributing everything. That is just nonsense. Under your eyes, people shouldn't be thinking differently from you.

Anyone thinking differently from you are downright wrong, and misattributing. What is the point of your philosophy? Forcing others to think the same as you do? That is not right.

Many would believe that your posts should be under proactive moderation for the low quality posts you have been spewing out with the meaningless quibblings stemming from your misunderstandings, and forcing people to believe and think exactly the same as yourself.

My posts are based on the philosophy of mathematics (Putnam)
— Corvus

Hilary Putnam?

How do your views square with indispensability? — TonesInDeepFreeze

Putnam edited a book called Philosophy of Mathematics Selected readings. He put in there various articles by different people. It is not a book solely written by Putnam. You obviously have no idea about the book, or what the Edited book means.

You lied about me when you said I started with insults. — TonesInDeepFreeze

If I really lied, then I would have told you that I lied, which is true. But you claim that I lied, which is false, and a lie.

I didn't lie, but you claim that I lied. Clearly and obviously you are telling a lie.
Therefore in whatever the case, you are the one who lied.

So, I am still baffled why you challenged me to cite a textbook when your own favorite book on set theory, which you claim to have read, is one of many many textbooks that give the definition you challenged me to show that it is in a textbook. — TonesInDeepFreeze

I was just telling you about Pinter's book to say that even classic Set theory books admit the historical controversies with the concept of infinity. I wasn't meaning to say the book is denying, accepting or defining on the infinity as per my view.

If you still don't understand what the point is, then you need to read a book called "The Oxford Handbook of Philosophy of Mathematics and Logic" Edited by Shapiro. Again in that book, there are various different articles with different views on the topic. But one that you must read about is "Quine".

The issue we've encountered is that the axiom of extensionality is simply false. — Metaphysician Undercover

The axiom of extensionality is

If A and B are sets, then A = B iff every element of A is also an element of B, and vice versa. — Open Logic

It tells us how to use the "=" sign. It is an instruction, and so is not the sort of thing that can be false. You either follow the instruction or you do not. If you do not follow the instruction you are not participating in the logic of sets.

The law of identity has various forms, but in set theory it is that

A=B iff both A⊆B and B⊆A. — Open Logic

This is a consequence of extensionality, not an axiom.

What Meta is doing is refusing to use "=" in the way the rest of us do. It's as if someone were to insist that the Rook move along a diagonal, all the while pretending that they had made a profound discovery about chess in doing so. Meta is simply not playing the game right.

To this sin Meta adds that of mischaracterising Wittgenstein. The private language argument is not that a symbol cannot be identical to an internal sensation, But that internal sensations cannot be treated in the way we treat other objects.

Under your eyes, people shouldn't be thinking differently from you. — Corvus

Well, one cannot play chess if there is disagreement as to the rules. A chess player expects their opponent not to think differently, at least in that regard.

You have made claims about the ideas espoused by various philosophers, but when challenged you have not produced citations or produced citations that do not support your claims.

You are not playing the game right.

And that is worth pointing out.

When I think about questions like 'what is mathematics really?' I tend to consider three different ways. How did mathematical skills arise in evolution? How do they develop during the lifetime of an organism? How could we make a machine that learns these skills 'without being told'? I won't say anything here about that third one.

Let's start with bees. Bees are capable of using numerical quantities in at least three different ways. Firstly, they can learn to recognise the number of objects that are present in a particular place. For example, they can learn to associate three objects with the presence of nectar, regardless of the shape, size, colour of the objects. They can also be trained to find their way around a simple maze where they have to learn to take the third turning on the left, for example. They can learn to do this even if the third turning is in different places. These are two different ways in which they can work with 'threeness': three things separated spatially or three things separated temporally. Bees can use oneness, twoness, threeness, fourness, fiveness, but things start to go wobbly there. Arguably they can use zeroness. Thirdly, they can use their waggle dance to communicate an approximate distance and direction. This is innate, inherited behaviour, and hence inflexible.

Next, some quotes from What Babies Know, ELIZABETH S. SPELKE

OBJECTS

... the movable bodies that we see, grasp, and act on. Before infants can reach for and manipulate objects, they organize perceptual arrays into bodies that are cohesive, bounded, solid, persisting, and movable on contact. Young infants use these abstract, interconnected properties to detect the boundaries of each object in a scene, to track objects over occlusion, and to infer their interactions with other objects.

PLACE

The core place system underlies our sense of where we are, where other things are, and what paths will take us from one place to another. Studies of animals and young children reveal that navigation depends, first and foremost, on representations of abstract geometric properties of the ground surface over which we travel: the distances and directions of its boundaries, ridges, cliffs, and crevices.

NUMBER

Research on human infants, children, adults in diverse cultures, and nonhuman animals all converges on evidence for an early-emerging ability to represent and combine numerical magnitudes with approximate, ratio- limited precision. This ability depends on a core system with most of the properties of the core object and place systems: it is present in newborn infants and functions throughout life, and it is ancient, unitary, and limited in the types of information it provides.

One might wonder at this point ask what it is that we've got that bees haven't. Perhaps they can't combine numbers. I don't think they have fully abstracted numbers from their environment. They can use threeness as a property in two different ways, but can they unify these notions of threeness? Could they be trained to take the nth turning after having seen n objects (for n <= 5)? That would be another step towards abstraction.

My own feeling is that for an agent to achieve full abstraction from its environment it needs to find some part of that environment where it can exert intricate control. A good way is making sequences of marks (or making rows of 'bodies that are cohesive, bounded, solid, persisting, and movable on contact'), and then looking at them. I think bees could make marks in wax and look at them easily enough, but I guess their environment does not give them sufficient motivation to do so.

Marks are made one after another in time in the sequence, but once made they are spatially separated. This helps unify notions of 'n-ness'. They persist in time, so extend memory capabilities. Sequences of marks can be created and modified by the agent, and by modeling this behaviour internally, the agent can make another step towards abstraction. The agent can start to predict what would happen if marks were modified this way or that. I would say that once an agent starts this sort of imagining, it has started thinking mathematically.

You have made claims about the ideas espoused by various philosophers, but when challenged you have not produced citations or produced citations that do not support your claims.

You are not playing the game right. — Banno

I have not made many claims quoting hundreds of philosophers. That is just another distortion of the truth with exaggeration. My point was simple, and I quoted one philosopher, from which was the Wittgenstein's writing, and mentioned 2-3 others. If you still cannot understand the point, you can look them up yourself, and find out. No one has to spoon feed you.

You are not playing the game right.

And that is worth pointing out. — Banno

I said this before, but will say again. Your problem is that you blindly say that others' points are wrong before presenting your arguments with evidence supporting your claims. That appears to be your trademark modus operandi of philosophy.

But because you keep on doing it firstly and unfairly to others, the other party will quite rightly try to argue against your wrong points and the style of your absurd claims dissecting the faults in your modus operandi. It is a vicious circle in your philosophy.

You have made claims about the ideas espoused by various philosophers, — Banno

I have not made many claims quoting hundreds of philosophers. That is just another distortion... — Corvus

:rofl:
indeed, it is. By you.

If you still cannot understand the point, you can look them up yourself, and find out. No one has to spoon feed you. — Corvus

I don't now actually recall what your point was. It wasn't very clear to start with, and is now buried in the clamour of your protest.

I don't now actually recall what your point was. It wasn't very clear to start with, and is now buried in the clamour of your protest. — Banno

See? That was what I meant. You don't even understand the point, but rubbish it as wrong. How absurd is that. By the way, you are still in deep illusion. I was not protesting on anything. I was just pointing out problems in your inaccurate posts.

Bees are capable of using numerical quantities in at least three different ways. — GrahamJ

I baulk at this. Bees only do bee things, and numbers are a people thing. I'd say that bees do things that people describe using numbers.

A small pedantry.

...which has me wondering if even you recall what your point was...

My point is that I agree that it is not the case that an abstract concept corresponds to one particular concrete instantiation, but rather we can only understand an abstract concept by thinking of some concrete instantiation of it — RussellA

That deserves consideration, though I'm not sure about it while also I don't have an argument in disagreement to give at this time.

What is a concrete example of the concept of 'does not exist'? What is a concrete example of 'there are things that do not exist irrespective of any list of properties that no existing thing have'?

Yet, the notion of 'concrete instantiation' is itself an abstraction made of the the two abstractions 'concrete' and 'instantiation'.

Also, the lines I'm thinking along is that certain utterly basic abstractions, such as 'object', 'thing', 'entity', 'is', and 'exists' themselves presuppose abstraction no matter what concretes are involved or not.

Anyway, to say that thinking of abstractions requires thinking of a concrete examples does not say that we don't think of abstract objects; or at least that to demand that we utterly eschew a notion of abstract objects brings even everyday conceptualization to a screeching halt or at least a devastating slowdown.

Putnam edited a book called Philosophy of Mathematics Selected readings. He put in there various articles by different people. — Corvus

When you said that you base on philosophy of mathematics and mentioned Putnam in particular, naturally I thought you meant that you base on views of Putnam. From what you said, one couldn't be expected to think that what you actually meant is that you have read a particular book edited by Putnam.

But now that you have added to your original statement, fair enough, your views are informed by reading that book.

So now, what are the articles in that book that you base your views on?

That book is full of great stuff. One article in particular that I think helps a lot is Boolos's 'The Iterative concept of set'.

You obviously have no idea about the book, or what the Edited book means. — Corvus

It is a really stupid inference from (1) I took you to mean that you base your views on Putnam when you said you base on philosophy of mathematics and mentioned Putnam in particular to (2) I don't know what an edited book is.

If I really lied, then I would have told you that I lied, which is true. — Corvus

That is false, since you didn't say that you lied but you did lie.

The plain record of the posts in this thread prove that you lied, as I explicitly linked to the posts. But you skip that.

I wasn't meaning to say the book is denying, accepting or defining on the infinity as per my view. — Corvus

You're confused as usual. I didn't say anything about your view of infinity regarding the book. Rather I note that you challenged me to show you a book in which 'infinite' is defined as I said it is defined in mathematics, while that definition is in the very book you mention as your reference. Again: It's not a matter of whether you agree or not with the definition, but rather that the definition is given in that book while you challenged me to cite such a book.

Meanwhile, my point stands that I think the chapter on the history and philosophy discusses the very point I made a while ago, but which you rejected, about the benefits of axiomatization.

one that you must read about is "Quine". — Corvus

I have read Quine. Not enough though, since logic and mathematics not topics to which I devote my primary attention. If there is something you have to say about Quine, then you can say it.

Anyone thinking differently from you (Banno) are downright wrong, and misattributing. — Corvus

Did Banno ever say or imply that he believes that?

Many would believe that your (Banno's) posts should be under proactive moderation — Corvus

Who are those many people?

The agent can start to predict what would happen if marks were modified this way or that. I would say that once an agent starts this sort of imagining, it has started thinking mathematically. — GrahamJ

That's an interesting idea.

I have not made many claims quoting hundreds of philosophers. That is just another distortion of the truth with exaggeration. — Corvus

The quote above, written to Banno, is exaggeration thus distortion.

Banno didn't say that you have made claims by quoting hundreds of philosophers..

My point was simple, and I quoted one philosopher, from which was the Wittgenstein's — Corvus

And that quote doesn't support the claim you made that mathematics regards 'infinite' as meaning 'finite'. Though, lately, you say that claim is only a metaphor for something. If you like, you may remind me what exactly you intend it to be a metaphor for.

Your problem is that you (Banno) blindly say that others' points are wrong before presenting your arguments with evidence supporting your claims — Corvus

He did not say that anyone is wrong merely by the fact of disagreement.

'1+1' does not equal '2'

'1+1' is not '2'

the denotation of '1+1' equals the denotation of '2'

the denotation of '1+1' is the denotation of '2'

1+1 equals 2

1+1 is 2

One may say that they don't themselves construe that way or that we shouldn't construe that way. But it is counterfactual to say that we don't construe that way.

/

One may claim that abstract objects should not be referred to in identity statements. But thought and communication, not just about mathematics, but even about everyday ideas, pretty much crashes if we are not permitted to apply the identity relation to abstract objects.

/

What passages in Aristotle are being referred to?

/

Saying that '1+1' names an operation rather than the result of the operation with the arguments is merely assertion. Again, one may say that one thinks that '1+1' should be regarded as a name of an operation, but that does not entail that in fact that is now how '1+1' is construed in ordinary mathematics.

/

The order of evaluation of terms and formulas is recursive. There is no ambiguity in that regard.

/

Yes, one can utter "1+1 = 8-1", but if the denotation of '+' is the addition operation on integers, the denotation of '1' is the integer one, the denotation of '8' is the integer eight, the denotation of '=' is the identity relation, and the denotation of '-' is the subtraction operation on integers, then "1+1 = 8-1" is false. This in no way vitiates that '=' stands for the identity relation. On the contrary, it is an example of the mathematics working just as we want it to work, just as it works even for the crank when he adds the number of pens on his desk, figures out his finances, relies on the entire body of science that allows him to stay alive, and uses the very computer he types on to send his ignorant, illogical and confused posts to this forum.

\

If one wishes to take '+' as intensional and not extenstional, then one should have a ball doing that. But that doesn't require that ordinary mathematics does that, espcially as, without some alternative rigorous framework, it would render thought and communication about even everyday mathematics unmanageable.

That's enough for now in reply to the star crank. It's just too laborious to correct every one of his confusions.

Several years ago, I moved to an apartment across town. Everything was very nice on the grounds of the apartment and in the neighborhood. Except the litter on the sidewalks. The first day, I picked up all the litter in front of my building, thinking that I wouldn't have to pick up for maybe another week or two. But the very next day, there was even more litter than I picked up the day before. And over time I started noticing that it was the same candy wrappers and soda cans every day. So it seemed that much of it was coming from a certain person. Someone daily spewing the same trash.

Crank: I want six plus two cans of that delicious abstraction-free metaphysical underground crank juice.

Shopkeeper: Here you go, eight cans of crank juice.

Crank: No, I said I want six plus two cans.

Shopkeeper. But six plus two is eight.

Crank: No, you must be reading those lying fool mathematicians with their extensional identity nonsense. I bet you even recite that awful axiom of extensionality every night before you go to bed. Six plus two means that I want you to get six cans then two more.

Shopkeeper: But I didn't have to do that. I already knew that I had eleven cans on the shelf, so I gave you all the cans on the shelf except three, so I subtracted three from eleven to give you eight cans.

Crank: No! No! No! I told, you just the way I posted at The Philosophy Forum, that six plus two is not eleven minus three. You're taking addition and subtraction extensionally! They are intensional, you dumb cluck! When I tell you what I want, I want it intensionally not extensionally! How can I be any clearer that six plus two is not eight?!

Shopkeeper: I'm very sorry, sir, but do you want the eight cans of crank juice or not?

Crank: Just forget about it. I want six plus two cans, not eight cans and definitely not eleven minus three cans! I guess I'll have to take my business elsewhere, even if I have to drive a potentially infinite number of miles to get there!