It seems to be used in place of "my perception" or "my recollection" which would be more correct usages. — Peter Gray

The ancient phrase "the truth, the whole truth, and nothing but the truth" refers to a subjective truth. So it's not a new trend.

There is the universal aspect, in that the function (meaning) of a word is to be used in a language game and there is the particular aspect, in that the meaning of a word is its use in the language game. — RussellA

What I had in mind was more like the difference between the dictionary definition of a word, and what the word actually means in any particular set of circumstances (context). So the dictionary definition states how the word is commonly used, in a general way, with a universal statement. But in any particular instance of usage, the context adds something, so that the use of the word is not exactly like what is described by the dictionary.

This indicates two very distinct interpretations of "meaning is use". One is to interpret that the meaning of the word is the universal, inductive conclusion of how the word is generally used, as outlined by a dictionary. The other is to interpret that the meaning is specific to each instance of use, and somewhat unique according to the peculiarities of the circumstances. The latter is the way that words are actually used. The former is very weak inductive reasoning.

I see trump is trying to rile up his racist followers to criticize the NFL. Maybe that explains why he randomly put out that Obama tape the other day. He was trying to get the ball rolling.

I would have thought that P1 “The meaning of a word is its use in the language” is quite central to Wittgenstein’s argument. — RussellA

I think you ought to consider that "use" has two principal meanings, one referring to the universal, the other the particular. So in the "use" of a tool like a hammer for example, there might be a universal rule, "a hammer is used for pounding nails", but this does not describe the particular instance of use, where it may be the case that "the hammer was used to crush a walnut". Wittgenstein recognizes these two very different and often conflicting aspects of meaning, what is given by the universal rule of "a practice", and what is given by the particular circumstances of practise. The former involves a sort of inductive reasoning, and the latter isn't necessarily consistent.

Wittgenstein doesn't say that rules are “necessarily external” in the sense of being outside a practice, outside a speaker, or imposed from above. What he denies is that a rule can be a private, inner object that fixes what counts as following a rule. — Sam26

Inside a practice, or a speaker would be internal to the mind of the person who performs the act. But the rule is something external to this, by Wittgenstein's definition. That is his very important "tool", this specific conception of "rule". The rule is external to the mind of the speaker, not something internal which the speaker follows in one's mind.

You basically say the same thing as me when you say " What he denies is that a rule can be a private, inner object that fixes what counts as following a rule". Therefore the rule must be external to the mind.

That’s also why the private language doesn’t need the premise “rules are external.” It needs the point that correctness requires more than a private impression of correctness. — Sam26

I cannot see the difference.

That’s also why the private language doesn’t need the premise “rules are external.” It needs the point that correctness requires more than a private impression of correctness. If you allow a “private rule” that has no criteria for correct reapplication, you haven’t saved rule following, you’ve emptied it. You can’t even make sense of “I’m following the rule” versus “I only think I am,” because there is no difference. — Sam26

That's exactly the point, and why rules are necessarily external, for Wittgenstein. If the rule was internal to the mind of the person, there would be nothing but "a private impression of correctness". Then there would be no difference between "the person is following the rule", and "the person believes that they are following the rule".

So this specific definition of what it means to follow a rule, positions the rule as external to the mind of the person speaking, and this move is a very important tool for Wittgenstein.

And the claim that “concepts are constructed with rules, therefore external” is too rigid. Language is full of normativity without explicit rules, and many concepts have family resemblance structure with flexibility. There are rules in the sense that some moves are correct and others aren’t, but that doesn’t mean the concept is a construction laid down by an external authority. The normativity is carried by the practice itself. — Sam26

Family resemblances are not concepts, they are a variety of different games. They evolve from free, unruly individual expressions, so they are not rule based. That's the point I made already. There is no rule as to what "game" means, no real concept of "game". However, Wittgenstein explains how one can apply boundaries for a specific purpose. To apply boundaries is to apply rules, and this creates a concept. Unruly, free evolution with family resemblances is clearly very different from concepts which are created by applying rules.

Wittgenstein’s tool is to block the fantasy that rules are private inner things/objects that determine their own application. — Sam26

So why would you deny that it is essential to Wittgenstein's position, a very important "tool", the proposition that rules are not inner things, but external to human minds?

I'm inclined to think that Wittgenstein was not concerned to refute the specific idea that pain is an object. — Ludwig V

In the book, right before he provided the example, with the symbol "S" to signify a sensation, he asked what would qualify as a criterion of identity for an internal sensation. And he used a chair as an example of an object with an identity.

I was impressed by the thought that if language is a system of communication, it is hard to see how it could not presuppose the existence of some sort of social relationship. So, at most, I was suggesting that a social context was a necessary condition for language. — Ludwig V

We can assume, and take for granted, that "language is a system of communication", but Wittgenstein was questioning whether this assumption is justified. That's why he proposed the possibility of a private language. If it is the case that language is dependent on rules, in the way that he defines rule following, then language is a system of communication. However, the family resemblances analogy shows that language might not be dependent on rules. That is why he had to go further in his analysis.

Third, the idea that “rules must be external” is too quick. — Sam26

For Wittgenstein, as stated in PI, rules are necessarily external. This is 'the essence of a rule' and it provides the basis for the distinction between "what seems right and what is right". It's a key premise to the so-called private language argument. If you allow the premise that a person could have a private internal rule, and judge oneself to be following that rule, the entire explanatory system of PI would be demolished.

In fact, I would say that this characterization of "rule" is a principal "tool" of Wittgenstein's. This is the means by which concepts, which are constructed with rules, are described as external instead of internal.

Are there whole new fields of mathematics just waiting to be discovered? Does this have any other possible impact on our daily lives? — EricH

I would say yes to the first question. There is clearly room for improvement on mathematical principles, therefore new fields waiting to be discovered. The second question is not quite so straight forward. But I would say, that unless you are one of the individuals who is going to forge those new fields, there is no immediate impact on your daily life.

I'm not clear what the difference is between a foundation and an ultimate foundation. But I don't see how inner feelings can be the only essential condition for language. They are necessary, perhaps, but not sufficient. If we were not social beings, there would be no language. Our form of life would be unrecognizable without inner feelings, social living, and language. — Ludwig V

I think this is an interesting way to put it. The necessary condition may not be sufficient, likewise though, the sufficient conditions may not be necessary. So "social being" turned out to be the sufficient condition for language development, but language could have developed under different sufficient conditions. Then the language which developed would not be the same as that which did develop, but would still be language anyway. "Sufficient conditions" may be tricky and difficult.

So for example, "social being" may be the result of language development rather than a primary condition for it. For example, it could be the case that when language first started to develop it was under sufficient conditions other than social being. Then, as language developed the condition of social being took priority and became the primary condition. I am not arguing that this is the case, only that "sufficient conditions" lack the necessity required to draw certain conclusions.

Are you saying that inner feelings exist independently of language? In an sense, that may well be true, but then social life can also exist independently of language. — Ludwig V

Yes, it's quite likely that feelings are prior to what we consider to be "language", and this would allow us to say that they would exist independently. However, by the difference between "necessary" and "sufficient" we cannot turn this around as you propose. What is apprehended as a "sufficient" condition is not necessarily prior, and can arise simultaneously.

Hi ,
I think we're actually very close to being in complete agreement here. We both place the inner feelings as prior, as "what make these language games possible", but there is some inconsistency between us as to how we interpret Wittgenstein's representation of "the concept". In the end, we seem to agree that concepts are not a part of the inner, but we both get there in slightly different ways.

The key to understanding the difference between us, I believe is to separate "inner" from "object". The inner is very real, but there is no such thing as an inner object for Wittgenstein. So, when "concept" is understood as abstract "object" it cannot be something inner.

Yes, a person can reflect on what they feel, but that reflection is optional and secondary. If you treat it as the foundation, you’ve already put the inner object picture back at the center. — Sam26

This is what Wittgenstein does, he puts the inner at the centre. His point though is that the inner is incorrectly portrayed as "object". This is what he demonstrates when he does the little thought experiment where he labels "a sensation" as "S", and ,marks in his journal every time that he feels "S". He is demonstrating that the recurrence of the inner sensation is not the recurrence of an object which can be named, as we name an external object. In this way he takes "object" out of the picture, but he leaves "the inner" as still central, but consisting of something other than objects.

You also say Wittgenstein rejects concepts, but that only works if concept means a private mental thing we consult before we speak. That isn’t Wittgenstein’s view. He relies on concepts in the public sense, the grammar of a word, what counts as using it correctly, what counts as a mistake, and what follows from it. If you deny concepts in that sense, you’re denying the very thing he’s investigating. — Sam26

What I said is that he rejects concepts as primary, fundamental, or natural. He describes the reality of concepts as a practise of applying boundaries for a purpose. So concepts are constructed with the use of language, the application of rules, formal logic, etc.. They are not something which underlie language use as its base.

Notice that "what counts as a mistake", requires rules, in the sense that it is an action contrary to a rule. But a "rule" under Wittgenstein's usage requires language for its existence. He is very clear on this. And "mistake" can only be judged as what is not consistent with conceptual rules.

However, the majority of natural language use (such as your example of "game") is not bound by these rules. Therefore the concept of "mistake" does not even apply to natural language use under Wittgenstein's description. When a person uses slang for example, using a word in a very unorthodox way, this is not a mistake. Neither is any new or innovative use of a word, a "mistake", because the person is not acting within a conceptual structure of "rules" concerning the words used. The person is creating one's own use within one's own mind, where rules do not apply.

The same point shows up in the game example. Wittgenstein isn’t saying there is no concept of game. He’s saying there’s no single essence of game. He uses game to point out that a concept can be held together by family resemblance rather than a strict definition. Saying “there is no concept” disregards his point and replaces it with something he never claims. — Sam26

Saying "there is no single essence of game" has the same effect as saying "there is no concept of game". To produce a concept of "game" requires rules which stipulate necessary criteria, an essence. Accordingly, there might be numerous concepts of "game" which people would produce for a variety of different reasons, but this is not saying that there is a concept of "game", it is saying that there is a multitude of concepts of "game".

I believe that his point is not to show "that a concept can be held together by family resemblance" it is to show a distinction between "concept" and "family resemblance". The natural way that meaning exists is as described by the family resemblance analogy. The concept however, is created by applying rules, boundaries which are applied for specific purposes. So the family resemblance usage of words may provide the basis for a multitude of different concepts of "game" ("game" as defined for this purpose and that purpose), produced from those different natural ways of using the word, but this is not a holding "the concept" together, it is a multitude of distinct concepts, each with its distinct set of rules. Following one set of rules would be making a mistake by another, and the same word, "game" supports distinct concepts. Notice specifically, that intentional ambiguity may be natural, and not a case of breaking any rules.

Finally, your picture collapses normativity into imitation. “Choosing to behave like others” explains copying, not rule following. Rule following requires the distinction between what seems right and what is right, between correct and incorrect moves. That distinction shows itself in training and correction. — Sam26

The point is, that rule following must be willful. We cannot force people to follow rules of language. So even training and correction require that underlying desire. Therefore "choosing to behave like others" does explain rule following. The fundamental desire for communion, to be a part of the group, is what enables rule following. Force does not enable rule following.

So, the point is simple. Inner feelings make these language games possible, but they don’t fix meaning. Concept isn’t some spooky inner tool, it’s the public grammar of use. And rules aren’t authoritarian commands; they’re the norms of what makes correctness and mistake intelligible. If you want to disagree with Wittgenstein, disagree with that, not with behaviorism or private mental classification, because those aren’t his positions. — Sam26

See, you are in complete agreement with me at the basic level, "Inner feelings make these language games possible. The "inner" is at the base of language. Where we disagree is with our understanding of "concept". What I'm saying is that Wittgenstein separates "concept", as we generally understand this word as an "abstract object", out from this "inner" which is at the base of language. But as you also agree, the "concept" is something dependent on community and language, therefore it is better described in that way, as a property of the public and its rules, rather than as something inner. So the point is that the inner is still prior and fundamental, as you agree, but "the concept" is not something inner, which you also agree to.

This is how we get to understand Wittgenstein's distinction between "what seems right and what is right". What is right is what is dictated by the rules and can only be judged in a public way. That is a grounding in justification. But, despite the fact that a person can learn and understand rules, and even apply them to oneself, the judgement will always be "what seems right" due to the influence of the inner, which cannot be rules, therefore never a proper "what is right".

But when “I’m in pain” is used as an avowal, or a cry, or a call for help, it’s a different language game. The grammar isn’t “I inspected an inner object and concluded,” it’s closer to “this is how we express pain, and this is how others respond.” — Sam26

I believe that this is a complex point, which should probably be looked at more closely. When someone says "I'm in pain", often they have inspected their inner feeling and conclude "pain" is the appropriate word. However, there are other ways that we respond to pain, reflexive recoil, "ow!", etc.. Notice that there is a difference, and Wittgenstein points this out.

The further issue is how others respond to the expression. "Pain" is a simple example, one does not need to shout "I'm in pain!", the "ow!" serves the same purpose. But the expression of many other feelings won't have the same sort of response from others, being less obvious. So this is why people tend to believe that "concepts" are at play here. They think there is specific criteria that a person applies when saying "I feel happy", or "I feel angry", so the person employs a concept before making that statement. This I believe is what Wittgenstein takes exception to.

I think that what he is pointing to is that in all of our expressions of feelings, we learn in the same way that we learn to say "ow!". Instead of analyzing the exact feeling, and deciding on the word which applies, we simply respond to the circumstances with the word that we've learned for that type of situation. And we learn by watching others. If something good happens, I say "I am happy", I don't analyze how I'm feeling, and decide that happy is the right word. When something bad happens I say I am said. Analyzing your feelings to apply a concept is what you learn from a psychologist.

Game is Wittgenstein's classic case. Board games, card games, children’s games, sports, video games, solitary games, competitive games, cooperative games. Some have winners and losers, some don’t. Some require skill, some are luck heavy. Some are played for fun, some for money, some as ritual. There’s no one trait that every game has. But there’s also no confusion in ordinary life. We learn the concept by learning a family of activities and how the word is used in each context or case. — Sam26

The point here is that there is no "concept" of game. And the larger point is that the idea of "concepts" is generally misleading. It's not a real description of how language works. A word gets a broad family of usage, but the word cannot be said to have a concept. However, we might attempt to create a concept by applying boundaries to usage. So the important point is that concepts are not a natural part of our language, serving to guide our word usage, they are something we attempt to create artificially by applying boundaries, rules of usage, criteria etc..

This points to the dichotomy indicated by @sime. The application of boundaries and rules, creates concepts, but this is not the natural way of language evolution. Thinking of "rules' in this way, as some top-down authority which the community holds over the individual, can be very misleading.

Look at the example above, of how we learn the expressions for feelings, like "ow!". By watching others, we learn how to express our feelings in specific situations. And we mimic what we've seen. You might call this 'learning a rule', but at the basic level, it's not a matter of community authority. Instead, it's a matter of choosing to behave in a way similar to others. That will, or desire to behave like the others, provides the "natural" base. Then, the training in boundaries, grammar, rules, and the formal education of '"concepts", is capable of taking advantage of this fundamental attitude, the desire to behave like the others.

So metaphysician undercover is now saying numbers are not ordinal, only cardinal. — Banno

Come on Banno, get with it, and quit your ridiculous straw manning. I'm saying there's no such thing as "numbers".

Unless you are referring to the symbols, what we call numerals, the assumption of "numbers" is blatant platonism, which we have no need for. We can still do orders, when 1, 2, 3, refer to first, second, third, etc..

Notice how this is much more realistic than the platonism you espouse. We get a clear distinction between "2" referring to a quantity, and "2nd" referring to an order position. This is simply a difference in the usage of the symbol. It's much clearer, easier to understand, and a more realistic representation, than the ambiguity of 'the number 2' being somehow both an order position, and a quantity.

That's the data from philosophers of mathematics. 43 respondents. Structuralism was ahead, with 18 agreeing. Platonism is int he alternatives, with 15 respondents.

Not perfect data, but far from a consensus for platonism. — Banno

The problem here has become obvious. there are probably many platonists, like Banno, who either don't realize it, or simply deny it. So response to a survey asking "what are you", would provide very misleading information. Better information would be obtained with some sort of questionnaire.

Quine's approach has a distinct advantage over your own, in that it allows us to do basic arithmetic. — Banno

Sorry, I don't see the relevance. We can do arithmetic without assuming that there are numbers which are positioned in between other numbers. All we need is symbols which represent values. Care to explain what you are trying to say?

Like I said, the ties with the CIA, MI6 and Mossad are clear. — Tzeentch

Let me tell you something about Epstein. Everything published about him which is "clear", is what he wanted the public to know. What he was really doing, his intentions, he obscured and kept secret. There are two sides to him, the public presentation, and the private, what he was actually doing. And, he was a master at secrecy. That's why he did what he did, so well.

The proffered alternative is that mathematical statements are true, and we can talk about mathematical objects existing, but this doesn't require positing some separate realm outside space and time where numbers "live." Instead, mathematical language works the way it does - we can truly say "there is a prime number between 7 and 11" - without needing to tell some grand metaphysical story about what makes this true. — Banno

Banno, the assumption that mathematical objects exist requires justification or else you're just talking through your hat. When anyone tries to justify their existence, Platonism is exposed in that attempt.

The truth of mathematical statements is connected to their role in our practices, proofs, and language games rather than correspondence to abstract objects in a Platonic heaven. — Banno

If your practise is to start with the premise that numerals refer to abstract objects, then the truth of this premise requires a platonic realm where these abstract objects exist. Otherwise any logic which follows is unsound, based in a false premise.

Here is the problem. For convenience sake, and common vernacular, we talk about numbers as if they are objects, and this in principle has no effect on mathematics, as mathematics is used. There is a clear separation between the talk about mathematics, people talking about numbers as objects, etc., and how the mathematicians are actually using the language of mathematics.

Describing mathematics in that way is just done to facilitate talk about mathematics. The talk about mathematics is in that way false, but it's a falsity of convenience, it facilitates our talk about mathematics. However, if the assumption that numbers are abstract objects makes its way into the axioms of mathematics (set theory), and this assumption is false, then we have a false premise within that logical system.

This view preserves mathematical realism (mathematical statements have objective truth values) while avoiding the metaphysical commitments of Platonism (no need for causally inert, spatiotemporally transcendent entities). — Banno

If it is the case, that within the axioms of mathematics, abstract objects are assumed, then "this view" which you present is a false description of mathematics. Clearly, set theory assumes within its axioms, abstract mathematical objects. Therefore the "objective truth" of mathematical set theory requires platonism.

You want to have it both ways (your cake and eat it, as frank says). You say that we can talk about numbers as abstract mathematical objects, though we know they really are not, and when we do mathematics the objective truth of mathematics is not dependent on this. That is fine in principle, if it is true. However, the truth about mathematics is that set theory assumes the existence of platonist objects, and the logical system is dependent on this assumption. This means that when we do mathematics using set theory, "abstract mathematical objects" is assumed, and the objective truth of mathematics is dependent on the "abstract mathematical objects".

So it is not just a matter of talking about numbers as mathematical objects, it is a matter of premising that numbers are platonic objects, and constructing a structure of mathematical logic with this premise as the foundation. That is set theory

Therefore, this talk about numbers as abstract objects, which we might recognize as false, yet still use, for simplicity sake, has been allowed to infiltrate and contaminate the system itself. We say that we recognize this assumption as not really a truth, but do we recognize the consequences? A vast logical structure, set theory, is based on what we recognize as a false assumption.

Platonism is not just "numbers exist", as Meta supposes. — Banno

Platonism is "numbers are objects". "Object" implies existing. When you propose that "X" stands for an object, or "2" stands for an object, the existence of that object must be justified. That's what Wittgenstein showed with the private language analogy. One can point to a chair, and say that is the object I'm talking about. But we can't point to a number this way. If I say that there is an object which is a number, this object must be independent from my mind, for its existence to be publicly justifiable, and that is platonism.

Otherwise the beetle in the box analogy applies. I have an object in my mind which I call "2", and you do too. We call them the same name, maybe even describe them in a very similar way, but your object is not the same as mine. therefore we do not have a proper "object" referred to with "2". The only way to justify 2 as an independent object is to assign platonic existence to it.

The response is not to reify the procedure that produces each digit; yet π is a quantified value within mathematics. It figures under quantifiers, enters inequalities, is bounded, approximated, compared, integrated over, etc. None of that is in dispute, and none of it commits us to Platonism. π is quantified intensionally, via its defining rules and inferential role — not extensionally, as a completed set of digits. — Banno

But you do not apply this principle infinite sequences. You do not say that each of these "is not a completed object. It is an instruction for producing digits". You insist on the very opposite, that these are completed objects That requires platonism to justify.

Go back and finish reading my post, instead of just replying to the second sentence.

If the rules of a single system contradict each other, as in the example, then "learning to use the rules" has a nuanced meaning, which includes choosing which of the opposing rules best suits one's purpose. Providing for an individual to choose from contradictory rules, according to one's purpose, allows subjectivity to contaminate the discipline which is supposed to provide for objective knowledge.

Both misunderstand mathematics, which consists in public techniques governed by rules. — Banno

That's insignificant drivel. We could say it about any discipline, they all consist of techniques governed by rules. That's education, learning the rules. The issue here however, is what do the rules say. If the rule says that "the natural numbers" refers to a completed object, that's platonism. If the rule says that "the natural numbers" refers to a count which can never be completed, then this refers to a mental act. The problem is that we cannot have both rules in the same system without contradiction within the system.

Depends on whether the first symbolism is time dependent. Does counting actually require temporal steps. Can you think of 1,2,3 as instantaneous? Just speculating. — jgill

This is the issue of platonism which Banno intentional avoids. The only way to believe that "N" could refer to a non-temporal, eternal object, is platonism.

What, you mean all the American billionaires and banks? — Tzeentch

American billionaires, and their banks are not a "state". Furthermore, you need to distinguish between the targets of the extortion and blackmailing, and the beneficiaries of the extortion and blackmailing.

You proposed that Epstein and his wife managed a "decades-spanning", "state-run enterprise" of "extortion and blackmailing".

If this is the case, then some state must have provided them with the funding, to operate, and in return that state would receive benefit from the extortion and blackmailing.

Have you seriously looked into which state was providing this funding, and benefitting from the operation?

Russia, ... really? :lol: — Tzeentch

Follow the money. Take a look at Epstein's funding.

Both Jeffrey and Ghislaine have deep ties to the CIA, MI6 and Mossad, leading me to believe this was a state-run enterprise. — Tzeentch

Which "state" would that be? The proposal of "state-run" requires that there is a specific "state" which runs it. The mention of association with a number of different states, is sort of contradictory to "state-run". And this is before even considering the possibility of Russian financing. To determine who "ran" the operation would require an understanding of who provided the funds.

Having ties to many states implies an entity acting outside the bounds of any state. Therefore it appears, at this time, like it was an operation run for personal gain, rather than for any state. That's why Trump appears to click right in to that clique. However, if it turns out that the financing was Russian, and it was "state-run", by Russia, that opens another can of worms.

I think you would approve of Wittgenstein's view. He was a finitist, and a math anti-realist. He didn't believe in set theory. He thought it was bullshit. — frank

I agree. Wittgenstein understood set theory is platonism, and rejected it as an inadequate representation of thinking. Thinking is the private property of subjects and is therefore inherently subjective. Platonism presents us with the products of thinking as something independent from the act of thinking, these are what we call "thoughts".

But this neglects a very important feature of thinking which is communication. We present our thoughts to each other through communication. When we allow that communication must be represented as a necessary aspect of "thoughts", then the true "objects" produced by thinking are the spoken and written symbols (leading toward nominalism), rather than some ideas which are called "thoughts".

The issue is that when we accept the reality, that communication is a necessary aspect of what is commonly called "thought", then it becomes very clear that the other representation of "thought" as some sort of ideas which are produced by thinking, has no grounding accept in platonism. The notion that thinking produces some sort of objects, ideas, is a misrepresentation, because what is actually produced is a system of symbols. And the existence of those supposed objects (ideas) have no grounding accept in platonism. So platonism is a false representation because it does not account for the role of symbols in the act of thinking.

Banno clearly takes the platonist perspective, which ignores the role of symbols, and we can see this from the following.

But this does not invalidate ZFC nor the axiom of choice, nor need we conclude that a limit is something the sequence approaches dynamically rather than a property of the sequence as a completed object.

And the larger point: At issue is whether there is one basic ontology for mathematics. Sime is seeking to replace one ontology with another, to insist that we should think of infinite sequences as processes or algorithms, not completed totalities. — Banno

We need to consider the role of symbols in representation, to understand the thinking which is being represented in these situations. Consider the following two ways to represent the natural numbers, "1, 2, 3, ...", and "N". Would you agree that these two symbolizations each signify something different? The latter represents a complete object which we know as 'the natural numbers". The former represents an endless sequence, which by that understanding, could never be complete.

With respect to those two distinct ways of representing "the natural numbers", would you agree that it is possible, even acceptable, and conventional, to represent what we know as "the natural numbers", in two contradictory ways? One symbolization means something, the other means something else, and the two contradict each other. This implies that there is two contradictory ways to understand what "the natural numbers" means, depending on the symbolization employed in usage.

@Srap Tasmaner.
This proposition, that what is meant by "the natural numbers" has contradictory meaning depending on the application, ought not be taken as an offence. You ought to accept it as a proposed description of the reality of mathematics, and judge honestly whether it is a true description or not.

And, the idea that an "object" within a highly specialized field of study like mathematics has contradictory definitions ought not be surprising to you. Take a look for example at the difference between rest mass, and relativistic mass in physics for example. The concept of "mass" has contradictory meaning depending on the application. This is just a description of the reality of human knowledge.

Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation. — Srap Tasmaner

OK, so you believe that mathematics is very much comparable to metaphysics, as I suggested. Do you also believe that to maintain consistency, if a philosopher believes that there is a need to be critical of metaphysical principles, that same philosopher ought to also believe that there is a need to be critical of mathematical principles?

When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject. — Srap Tasmaner

Your imagination misleads you then.

So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations. — Srap Tasmaner

I don't understand this feeling of offense. This is philosophy, and what we do is critical thinking, and therefore criticize. What I don't get, is that many people think it's acceptable, even warranted and expected, that we criticize metaphysical principles, yet some of the same people believe it's for some reason unacceptable, and offensive to criticize mathematical principles. Where is the consistency in this type of attitude?

What I apprehend here is that some people take mathematics as a sort of religion. So in the same way that some people get seriously offended when their "God" is criticized, some others get seriously offended when their "mathematics" is criticized.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm. — sime

This is what @Banno seems to be in denial of. The intent behind creating the infinite sequence, is to create an infinite sequence. This implies that the so-called "limit", as defined by Banno, is prior to the sequence, as a requirement for the creation of the sequence.

On the other hand, we could look at the infinite extension of pi, as an unintentional infinite sequence. Notice, that now there is no "limit". This exposes the nature of "the limit", it is a concept which serves the purpose of creating an infinite sequence. When an infinite sequence is created unintentionally, there is no "limit".

This leads to a question about the intentionality of the infinity which is the natural numbers. If this is an unintentional infinite sequence, we ought to assume that there is no limit. But if it is intentional, then there ought to be some sort of limit, as the source of its creation.

Your denial never ceases to amaze me.

This is exactly arse about. The limit is a result of the sequence. Those who care to look can see exactly that in the proofs offered earlier. — Banno

Take a look at the quoted sequence:

he key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1 — Banno

Quite clearly, the limit must be assumed prior to taking half of it.

look at what "as close as we like to some number" means. It means there is no limit to how close we can get to that number. That is how you define "limit" a specified number for which there is no limit to how close we can get to it.

Added: the pedagogic problem - it's not a mathematical problem - is how to dissipate the notion that the limit is "a little bit more" than the sequence? — Banno

Obviously, there is always "a little but more" in terms of how close we can get to the limit. that is implied by your definition of "limit". If we'd like to get closer to the limit, than any previously proposed closeness, we can do that, and get closer to that limit. This is what your definition indicates. Therefore, to "dissipate the notion" that there is always more, would be a big mistake, contrary to the definition. Why would you aspire to do this?.

Notice that the limit is set out in terms of the sequence - the limit is provided by the sequence alone! so the limit results form the sequence. But it need not be one of the elements of the sequence. It's not something the sequence reaches toward — it is a property of the sequence itself. — Banno

So here is where your mistake lies. The limit is the condition for the sequence, the sequence is derived from it, as a formula, a repetition of "half the value between this point and the specified limit". Therefore your mistake is in saying "the limit results form the sequence". The limit is necessarily specified prior to producing the sequence. Then, the sequence is produced from the way that "limit" was defined. We can always get closer to the limit, if that is what we want to do.

Because the limit is prior to the sequence it is not "a property of the sequence itself". The limit preexists the sequence as a necessary condition for it. So if one is to be said to be a property of the other, the sequence is a property of the limit. By this mistake, what you say which follows, is all wrong.

The limit isn't something the sequence is trying to get to; it's a concise description of how the sequence behaves. The sequence doesn't "know about" or "aim for" its limit - the limit is simply our label for a pattern in the sequence's terms. — Banno

The sequence is designed, and produced from the limit. Therefore knowing the limit, and aiming for it, in this way of getting ever closer to it, is an essential aspect of the sequence. the sequence is derived from the nature of "the limit". And this is clear from the way you define "limit". We know "some number", and we also know that there is no limit to how close we can get (we can get as close as we like) to it. The sequence is derived from the specified number.

If ∣x∣<ε for every ε>0, then x=0 is not a stipulation about limits; it is a theorem about the real numbers, derived from the order structure of ℝ. — Banno

The conclusion "x=0" is not valid without a further stipulation that there can be nothing between the least ε and zero. But we know there is no least ε and there will always be another lesser ε . Therefore x has no place in that number system, and is wrongly inserted as a category mistake. What is x? And how is it allowed to fit into the number line in this way, when it is not itself a number?.

Since what constitutes "the real numbers" is a matter of stipulation, you are wrong to say it's not a stipulation. You have inserted, through a category mistake, something called x which is not a number, but somehow you claim that it is equal to a number, zero in this case. That is a stipulation.

I'm not concerned about credibility or showing that I'm working. — frank

Banno's proofs continue to be a matter of begging the question. Stipulate that the limit is the value, then use that as a premise in proving an instance of this.

The electron is, in fact, conceived by scientists as a point. It's startling, but true. — frank

That's half true, because the electron is also conceived as a probability cloud. Hence the wave/particle duality.

But the point I made is that "point particle" is a conception of convenience, designed for the purpose of representing interactions. It does not represent how the electron is actually conceived as existing. The electron is modeled as a "point particle", but it does not exist that way, the probability cloud is a better representation (though still very inadequate) of how electrons exist.

For (2) to be possible, I must be offering you the actual value. — Srap Tasmaner

Sorry Srap, I can't see how you make this conclusion. 'Within a specified tolerance' does not indicate "the actual value" has been given. It just indicates that the value is within a specified tolerance. In neither case is the value which is being rounded off, actually specified. If "the actual value" was specified the procedure would be unnecessary. So I don't see any significant difference between the two, just two different forms of rounding off.

But an electron is conceived as a point. — frank

I don't think so, electrons are conceived as a cloud of probability, with a variable density.

Isn't that the same as the idea of an infinitesimal in math? — frank

An electron could not be infinitely small, because this would reduce the probability of them having any location to practically zero. And that is contrary to what is observed and verified by the cloud of probability conception.

The issue is actually quite complex, because "point particle" is really just a conception of convenience. It's not meant to actually indicate the physical properties of the supposed particle. Rather it's a convenient way to conceptualize interactions. Compare this to the concept of "centre of gravity" for example. This is meant to represent a point which indicates where a body's weight or mass is centred around. But it's just a conception of convenience which helps to model interactions, it doesn't indicate a real point that the body is centring itself around. Nor does the "point particle" concept indicate a real point where an electron is located. They are both conceptions of convenience, intended to facilitate the representation of interactions.

According to Zvi Rosen, the sum and the limit are not equal (according to Cauchy). They're just as close as we "want" them to be. — frank

It's just a matter of definition. Notice what you say, that they are as close as we want them to be. Banno wants them to be equal, and so he defines them that way. But in the context of this discussion such a stipulation is really meaningless.

The salient bit today is that a limit is not a rounding off. — Banno

Then why did you say to@jgill, "a more intricate form of 'rounding off'"? That really looks like "rounding off" to me. The point being, that applying a limit to that which is limitless (infinite), is nothing other than a form of rounding off. it's really no different from saying that pi is 3.14, or that it is 3.14159, or however you want to round it off. You are apply a limit to what is limitless, and that is a form of rounding off.

It's not that the adjacent members of a sequence become "infinitely close": they become "arbitrarily close", and so the series (in this case, the sum of the members of the sequence) becomes arbitrarily close to — well, that's the thing, to what? And that's your limit. — Srap Tasmaner

That's exactly when rounding off is employed, when things are designated as "arbitrarily close". How have you done anything other than described a case of rounding off?

The difference between the limit and the sum is an infinitely small number. — frank

"Infinitely small number" really has no meaning in this context. If the formula is applied to spatial distance, as in the Zeno paradox, it means infinitely short distance, not infinitely small number.

We could say that this solves Zeno's paradox as along as space and time actually conform to the calculus framework. I think the average scientist would agree that they do conform, but there is still room to reject the calculus angle. — frank

I don't agree. I think the average scientist would say that it doesn't make sense to talk about infinitely short distances. So if they round something off to zero it wouldn't be an infinitely short distance which is being rounded off, because the limitation of practise would require rounding off before infinitely short distance (whatever that actually means) is reached.

For example, when I use pi I round off to 3.14. Some scientific applications might request something more precise, but really the precision of the outcome is relative to the precision of the actual measurement. But, it's never an infinitely short amount which is being rounded of. So in the other example, are the measurements such that you are rounding 1/2+1/4+1/8 +1/16 to 1, or are you rounding 1/2+1/4+1/8+1/16+1/32+1/64 to 1? In the first case, 1/16 would be lost, as rounded to zero. In the second case 1/32 would be lost. The smaller the size becomes, the more difficult it becomes to measure it, and the required precision is application dependent.

I'll repeat, since you did not address the issue.

It is a difference between theory and practise. In theory, the sum approaches the limit. In practice the sum is the limit. The latter can be understood as "rounding off". Failure to recognize this is to misunderstand.

The key is that an infinite sequence may have a finite sum: ½ + ¼ + ⅛ ... = 1 — Banno

You mean the key is to put an end to the infinite sequence by rounding off. That's what we've done with pi for thousands of years. But if you think that this puts an end to the infinite sequence, and solves Zeno's paradoxes, you misunderstand.

This obviously works in practise. But the paradoxes are theoretical, they always have been, and they've always been irrelevant to practise. "Limits, infinitesimals and calculus" change the practise, but have no affect toward answering the paradoxes, which remain unchanged despite the changes in practise.

Replayed songs are physical — Corvus

isn't the firing of neurons, which constitutes the playing of the song in the mind, something physical as well? It doesn't happen at the speed of light, because it occurs through a physical medium. So wouldn't time dilation slow down that activity?

But you cannot access the other folks mind, hence you wouldn't know what song is being played in his/her mind. — Corvus

Consider, you could have an analyzing system hooked up to the person's brain. The person tells you i am playing Social Distortion's "I was Wrong" in my mind, and you observe the corresponding neural activity. Then, whenever you see an exact replication of that physical activity you know the person plays that song. From a different frame of reference, would time dilation apply? You might see the same activity slowed down.

No. — Corvus

Why not? it's like when you play a 45 at 33 1/3.

Does that explain it? — frank

No, you described a long process, and the problem is with the use of "at some point". How does a process occur at a point?

Ok. All I know is that it's common sense that if you're driving from Washington DC to Alaska, you will, at some point, be in British Columbia. Those who claim this view is wrong should at least acknowledge that what they're saying sounds bizarre. — frank

I would say the opposite is the case, what you say sounds bizarre. You are representing driving through British Columbia, as being in British Columbia at some point. What does "at some point" even mean in this context? You use it because it's an acceptable figure of speech, but taken literally, it doesn't fit. So what does it really mean?

Trump is being fetishized as the personification of pure evil... — Tzeentch

From the "tribal" perspective, that which divides the tribe is evil. And the designation is justified.

Maybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner

When the measuring stick needs to be measured, it's time to throw away the measuring system completely, and devise anew. Otherwise paradoxes are produced, like Russell's.

Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. — Srap Tasmaner

Of course, an infinite collection by any standard definition of "collection" is nonsense. A collection consists of things which have been collected, not things designated as collectible. And that's problems arise in set theory, "collection" becomes a designated collectible type, rather than the collection itself.

This is how the concept of "the empty set" creates paradoxes like Russel's. A "collection" with no items is not a collection at all. It is only a criterion for collection, therefore an abstract 'type" distinct from an item. Allowing for an empty set means that "the set" itself is not the collection of things (or else an empty set would not be a set), but "the set" is the abstract type, which describes the things to be collected. The things themselves, therefore, the elements of the set, must be categorically distinct from the sets, or else the empty set is the contradictory notion of a collection of nothing. Failing to follow this categorical distinction, which is necessitated by "the empty set", and allowing that a set might itself be an element of a set, produces problems.

But if the collection consist of things designated as collectible, and there is none of them, then it makes sense to talk about an empty set. However, this leaves cardinality as completely unjustified because the elements are just possibly collected, and therefore not counted.

Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? — frank

I think that what Srap is saying is that we cannot reduce motion to a succession of truths. That's what Aristotle demonstrated as the incompatibility between being and becoming. If change is represented as a succession of different states of being, one after the other, then there will always be the need to posit a further distinct state, in between any two. Then we have an infinite regress, without ever accounting for what happens between two states, as the change, or "becoming" which occurs as the transition from one to the next.

So if motion is described as getting from A to B, A and B are the two points of being, you are at A, then you are at B. Since they are not the same, there is distance between, and we can posit a middle point. You are at C. Then we posit a point of being in between A and C. You are at D. Notice, we've reduced motion to "being at a point which is different from the previous point". But this produces an infinite regress without ever addressing the real issue of how you get from one point to the other, what happens in between. This is the real nature of motion, what happens in between, and it cannot be represented as being at a designated point.