My understanding of Wittgenstein's idea that meaning is use is that a word's meaning is not given by a definition which purportedly captures the word's essence but by the context in which it appears. — TheMadFool

A better way to approach it is to forget about meaning and look to use. Knowing what a number is consists in being able to count, to add, to subtract, to do the things that we do with numbers; not with a definition set out in words.

Wittgenstein wrote much regarding philosophy of mathematics, and considered it is more important work.

I appreciate your making an effort to understand my argument, and largely succeeding. I would like to invite and and others to try to state my argument in their own words ("steelman" my argument).

I don't know how intuition and "psychological well-being and development" are related but we do feel upbeat when our intuition is right on the money, hits the bullseye — TheMadFool

This is not what I meant. Let me try to summarize my argument. To see what I meant, please see my 5. below.

Perhaps this is related to your attempt to link intuition with "psychological well-being and development" — TheMadFool

Please see my 5. below:

1. "2 + 2 = 4" (with or without 's "cyclic group" qualifier) cannot ultimately be known. Its knowledge ultimately rests on a feeling of knowing, which is a kind of intuition, and intuitions are not objective [Edit: objectively] [Edit: 100%] reliable as justifications for knowledge (because, for instance,

The fact that we have near-zero knowledge of intuition is a major stumbling block in advocating it as a reliable technique for problem solving. — TheMadFool

). Intuitions can become more and more correct, however (see my 4. below).

2. My 1. above is trivial in one sense, the sense that admitting it should not cause us to hesitate for a moment to rely on 2 + 2 = 4 when we're planning a landing of some kind on Mars.

3. There is another sense in which my 1. above is not trivial – the sense that admitting it motivates us to want to know practically how to avoid/prevent occurrence of the "feeling of knowing" neurological event when we are contemplating 2 + 2 = 5, and thus may lead us to learn how to avoid/prevent such occurrence. This is of epistemological significance.

4. There is another sense also in which my 1. above is not trivial – the sense that admitting it motivates us to want to improve the reliability of our feelings of knowing. I think that we all, to different degrees, possess deep in our minds a capacity for more and more accurate feelings of knowing – a capacity that I would say, in line with Aristotle, is a combination of innate and learned. This is of epistemological significance.

5. Introspecting (initially prompted by trying to understand our intuitions, as by whatever prompts it) can lead to greater psychological well-being and development. This is not, or not entirely, of epistemological significance. Let’s call it fringe benefits of the quest for more correct intuitions.

As a brief argument in support of my 4. and 5. above, I would say that introspecting, particularly through a regular practice of meditation, serves to throw sunshine, the best disinfectant, on kinds of psychological clutter that interfere with the free, efficient, and thus also healthy movement of mental energy. Just one example of psychological clutter would be an emotional investment in the correctness of some political or academic ideology. The removal of psychological clutter allows us to access deeper levels of our minds than we had accessed before, a kind of development. The deeper levels, besides being repositories of more correct intuitions, are generally more peaceful and more characterized by a sense of radiance.

A better way to approach it is to forget about meaning and look to use. Knowing what a number is consists in being able to count, to add, to subtract, to do the things that we do with numbers; not with a definition set out in words.

Wittgenstein wrote much regarding philosophy of mathematics, and considered it is more important work. — Banno

It does seem quite within the bounds of reason to "forget about meaning and look to use"; after all, humans and some animals like bonobos, ravens, etc. are seen as toolmakers and being so one entry in our list of priorities would be versatility in our tools. It's likely that we're more interested in how something, including words (and numbers), can be used rather than what they mean. Perhaps numbers too have uses outside their natural environment (mathematics) and can be put into service for other purposes for which they're, by a stroke of luck, well-suited for. Can you think of a non-mathematical use for numbers? I'll give it a shot, 666!

I don't see how we can escape from the essential role of a pattern of synaptic firings that results in a subjective feeling of knowing. And then the problem is, as I suggested earlier, that if one day my brain functions differently than it usually does, that pattern might be triggered not by 2 + 2 = 4, but by 2 + 2 = 5. Evolution has guaranteed that such days will be rare, but is a high order of probability the best we can do in trying to prove that 2 + 2 = 4?

If I'm missing something, I hope that someone can pinpoint what that is. — Acyutananda

You are correct, strictly speaking. Practically speaking, this does not apply to arithmetic anymore, unless you were a raised as a feral child, a.k.a Mowgli style. The formalization of such extremely rudimentary and materially manifest abstractions doesn't happen under spontaneous impetus. Those ideas were internalized, starting long ago, with routine behavior associations in our remote animal ancestors, as @Banno proposed them to be, then they were gradually absorbed into awareness through notions that articulate vaguely aspects of nature, and then finally conceptualized. Conceptualization also follows a historical process of refinement involving the civilizational fabric of society and the formal academic convention, passing through stages of eccentricity that resemble arithmetical theism. So, your spontaneous conception of ideas regarding the basic qualities of nature, such as arithmetic, are relatively unimpactful, because you are entrenched into continuous multi-generational collective refinement of those concepts, spanning many evolutionary stages.

Theoretically it could, but in reality, it more so applies to the axiom of choice or law of excluded middle. We are not sure where to look for correspondence to those axioms. The methodology that should determine their soundness is debatable. The problem there is different in some sense. Because the experience that needs to arbiter the design of our abstraction is not immediate and obvious. So, we are left at the mercy of guesses, but not for materially manifest pervasive aspects of nature, such as quantification. Even if you are theoretically correct, that the collective solution can still be wrong, the chance for it is much smaller then the odds of perishing in an ecological catastrophe in the next decade.

"2 + 2 = 4" (with or without ↪simeonz
's "cyclic group" qualifier) cannot ultimately be known. Its knowledge ultimately rests on a feeling of knowing, which is a kind of intuition, and intuitions are not objective reliable as justifications for knowledge — Acyutananda

I have a discussion with another member of this forum in a different thread. I am arguing there that ultimately everything rests on innate conviction, or persuasion, and that it cannot be denied. Of course, one can continuously reevaluate the quality of such persuasion, as they gain new insight and amalgamate their various persuasions, but again, even if a person is wrong about something, one can always hope that nature will decrease their chance of thriving and evolution will replace their erroneous influence. So, don't worry about it. Genocide is a form of logical argument.

P.S. : I am a little out of line here, but I hope you understand my well meant drift. You don't have to be right.

Thanks for these two posts, and I may get back later to these and some points in your earlier posts.

To all: Sorry to have written so much already, but, does anyone here know of any formal papers that argue that propositions such as "'2 + 2 = 4' cannot be proved, but rather rests on intuition" and "'A square must be rectangular' cannot be proved, but rather rests on intuition" are correct, but trivial?

Can you think of a non-mathematical use for numbers? — TheMadFool

I don't see the relevance...

I don't see the relevance... — Banno

You have to be kidding me. If meaning is use, we should be able to use "2" in some way different to what it was intended for (counting) or, if we should forget about meaning and look to use, it must be possible to use "2" in any way we want. I ask you to summon your powers of imagination and use "2" in a way that doesn't have anything to do with counting.

if we should forget about meaning and look to use, it must be possible to use "2" in any way we want. — TheMadFool

We can.

We can. — Banno

An example?

1. "2 + 2 = 4" (with or without ↪simeonz's "cyclic group" qualifier) cannot ultimately be known. Its knowledge ultimately rests on a feeling of knowing, which is a kind of intuition, and intuitions are not objective [Edit: objectively] reliable as justifications for knowledge (because, for instance — Acyutananda

Intuitions can become more and more correct, — Acyutananda

How did you come to the conclusion that "Intuitions can become more and more correct" when you know that "intuitions are not objective [Edit: objectively] reliable"?

The way it seems to me, the two statements made by you (above) don't jibe.

There is another sense in which my 1. above is not trivial – the sense that admitting it motivates us to want to know practically how to avoid/prevent occurrence of the "feeling of knowing" neurological event when we are contemplating 2 + 2 = 5, and thus may lead us to learn how to avoid/prevent such occurrence. This is of epistemological significance — Acyutananda

Are you referring to wrong intuitions when you say, "the feeling of knowing neurological event when we are contemplating 2 + 2 = 5"? Well, I did touch on that and it's precisely because there are such "events" that we should be careful about relying on intuitions. Just so you know, 2 + 2 = 5 isn't always incorrect if you take into account, for instance, the fact that "5" is an arbitrary symbol and can be used, if we want, to symbolize the quantity |||| (four).

Perhaps, if you give this some more thought, intuition could be a mental process that arises from a deep understanding of how the mind works - its habits, its propensities, the capabilities and limitations of its constructs, its default states, its rhythms, its objectives - and that's why what we think are "wrong" intuitions may actually reveal very profound truths about the human mind.

I recall reading an elementary book on math for teachers and every chapter in it has a section on mistakes - what to expect, why children make such mistakes, what such mistakes reveal about a child's mind, and so on - and I think a similar approach should be used in studying "wrong" intuitions and even right ones too for unbeknownst to the conscious mind which works within mental constructs like logic and theoretical frameworks, the unconscious (intuitive) mind may actually be, in a sense, operating at the outer limits of such mind creations and "wrong" intuitions may reveal, in a manner of speaking, how we think rather than provide information on what it is that we're thinking about.

I think I'm off-topic but I had to share my own intuitions about intuitions with high hopes that it might shed some light on your concerns.

Just a partial answer for now:

How did you come to the conclusion that "Intuitions can become more and more correct" when you know that "intuitions are not objective [Edit: objectively] reliable"?

The way it seems to me, the two statements made by you (above) don't jibe. — TheMadFool

Okay, I have now edited one of those sentences again. Now it is:

"intuitions are not objective [Edit: objectively] [Edit: 100%] reliable"

That is, "intuitions are not objectively 100% reliable."

I think that a person who holds within themselves an incorrect intuition can eventually find in themselves a more correct intuition on the same topic, and later a still more correct intuition.

I doubt that the person can ever find in themselves a 100% correct intuition, but who knows. Maybe the Buddha had intuitions, or at least moral intuitions, that were 100% correct.

What I'm trying to tell you is there may not be such a thing as a wrong intuition and therefore there's no "progression" such as that from 0% correct to 100% correct intuition. All intuitions, according to my "theory", are correct and reveal different aspects of an issue. A moral intuition, since you seem concerned about it, could shed light on different levels of thinking - from the behavior of individual atoms in a single neuron when one encounters the words "moral", "immoral", and "amoral" to complete, fully-operational, moral theories that could, in principle, being cosmic in scale.

The meaning of "2" is not set out in a definition, but seen in what we do with numbers. Meaning as use. — Banno

So then 2 + 2 can equal 22

We can. — Banno


An example? — TheMadFool

May I?

"Would you care for another glass of 'Two Barrels'*?"

*it's a brand of Whiskey.

You have to be kidding me. If meaning is use, we should be able to use "2" in some way different to what it was intended for (counting) or, if we should forget about meaning and look to use, it must be possible to use "2" in any way we want. — TheMadFool

"2" could be code for my lawnmower. We do look to use to discern meaning. The notion that meaning is identical to use is wrong. Without some predetermined meaning, symbols can't be used for anything.

"Would you care for another glass of 'Two Barrels'*?" — Isaac

That's a play on words, both meanings are using "two" as a number.

An obvious non numerical example: dynamic programming languages tend to treat '+' as a concatenation operator when applied to sequences of 'chars'. So x = '2' + '2'; means that x is equal to (or has the value of) the string '22'. Not the integer value 22. This also answers @ArguingWAristotleTiff yes 2+2 can equal 22. If + represents concatenation.

This is confusing syntax and semantics. Strings are infinitely dimensional space, we are talking about the concept of integers, which are ordered set.

Strings are infinitely dimensional space — simeonz

I have no idea what that means.

What I mean is, that if you try to create a basis, x_1, x_2, ..., such that each string can be represented as a sum of i_1 * x_1 + i_2 * x_2 ..., you will need infinite number of x's. Not to mention that the operation is not commutative. This is not the same concept. It does not follow the same algebraic rules. It is a counter-example of arithmetic.

Strings are infinitely dimensional space
— simeonz

I have no idea what that means. — emancipate

:rofl:

That is, "intuitions are not objectively 100% reliable." — Acyutananda

Simple materially implied intuitions can become very reliable. When they have been ratified from experience for generations and convention has reached consensus, there aren't a lot of variables left in their definition that provoke further refinement. That is why mathematics focuses on simple pervasive intuitions and builds the rest from them. This is what distinguishes it from physical sciences that are much more susceptible to constant amendment.

I should mention, we don't need abstractions to match the world exactly. We don't expect them to. We just need them to match it sufficiently to be useful. As I said, approximately, probabilistically, homomorphically. Any actual ball in the physical world is not precisely spherical. No matter the interpretation, the contact surface of a basketball is not exactly equal to 4 pi r^2, where r is half the longest distance between two atoms of the ball. The reason the interpretation is irrelevant, is because no two basketballs are exactly the same, and hence the formula could never be accurate for all of them. But, with some latitude and sense of utility, this is good enough approximation that captures much of the character of the real objects using a significantly simpler description. We are not trying to turn the objects into literal mental images, but just to handle them efficiently.

I was attempting to give an example of a non-arithmetical use of the symbol '2'.

It is a counter-example of arithmetic. — simeonz

So my attempt was successful(?).

"2" could be code for my lawnmower. We do look to use to discern meaning. The notion that meaning is identical to use is wrong. Without some predetermined meaning, symbols can't be used for anything. — frank

I don't see why a symbol must have "some predetermined meaning". The algorithms for assigning meaning to a symbol and vice versa is as follows:

Algorithm A [When the symbol precedes meaning]
Step 1. Invent a symbol e.g. [imagine yourself coming up with a brand new symbol]

Step 2. Assign a meaning to the symbol you invented

Algorithm B [When meaning precedes the symbol]
Step 1. Meaning in need of a symbol because as when language first begins e.g. [that thing that flows down the mountain side, you can quench your thirst with it, it's transparent, etc.]

Step 2. Invent a symbol [water]

Step 3. Assign the meaning under consideration to the symbol

Plus, a symbol maybe given any meaning one wants: "2" could be code for my lawnmower and I have this feeling that language is, at the end of the day, code and the symbol-meaning relationship is completely arbitrary i.e. we have unlimited freedom as to what a symbol's meaning can be. Try it. Oh! You already have.

May I?

"Would you care for another glass of 'Two Barrels'*?"

*it's a brand of Whiskey. — Isaac

Great but "Barrels" spoils the show in a manner of speaking.

So my attempt was successful. — emancipate

The problem is, that you demonstrated that syntax could be abused, not that concepts with strict semantics behind the syntax can be used in innovative ways. Strings can actually be ordered lexicographically, but are something called free monoid, and their ordering is not well-ordering. Natural numers are sigularly generated (by the successor relation) commutative monoids and are well-ordered.

Yes that is true. My example is at the syntactical level. Ok I'll leave it to someone else to find an example at the semantic level.

I was just saying there are some prerequisites for language use. Some of it is probably innate capacity, some of it is learned.

Meaning doesn't spring into being in a unique case of language use like Venus out of the ocean.

An obvious non numerical example: dynamic programming languages tend to treat '+' as a concatenation operator when applied to sequences of 'chars'. So x = '2' + '2'; means that x is equal to (or has the value of) the string '22'. Not the integer value 22 — emancipate

:up: