2 + 2 = 4

I I [2] + I I [2] = I I I I [4]

:joke:

I thought we're at liberty to define mathematical operations to suit our needs. There's nothing true/false about a definition, it just is the way we decide it should be.

Also, to my knowledge, intuition isn't "rejected" outright in Western philosophy. It's validity as a useful tool in solving problems [what else is there for us to do? :sigh: ] is widely acknowledged even though we're completely in the dark about how intuition works. 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. Nevertheless, some people seem to have a knack for it or perhaps it's just blood, sweat, and tears masquerading as intuition.

I I [2] + I I [2] = I I I I [4] — TheMadFool

I = ?

I I [2] + I I [2] = I I I I [4] — TheMadFool

I would put exactly the same but as fingers. 2 plus 2 equals 4 because we literally count it with our hands. I guess this is the best proof.

.

I = ? — Tzeentch

I guess @TheMadFool referred as sticks but it also works with counting with your own fingers. This is the most solid proof of why 2 + 2 equals 4.

I guess TheMadFool referred as sticks but it also works with counting with your own fingers. This is the most solid proof of why 2 + 2 equals 4. — javi2541997

Sticks, fingers, luxury yachts. They don't answer the question of what is one.

2 + 2 = 4, this is if quantity does not turn into quality. A glass of sugar + a glass of boiling water = 1.5 cups of syrup, not two. :)

I = ? — Tzeentch

I = 1 of course.

I would put exactly the same but as fingers. 2 plus 2 equals 4 because we literally count it with our hands. I guess this is the best proof. — javi2541997

:up:

They don't answer the question of what is one. — Tzeentch

I guess they do. Because the question is why 2 + 2 is 4 when supposedly it is not proven. He said is just “trivial” but if you count your fingers, stickers and luxury yachts you will see why we end up in 4 because is basic reasoning

He said is just “trivial” but if you count your fingers, stickers and luxury yachts you will sew why we end up in 4 because is basic reasoning — javi2541997

2 threads + 2 threads = 4 threads, exactly? Not 1 river?

Because until now people don't believe you can build a society based on intuition. And thís is exactly the difference between science and philosophy. Because scientists don't argue about whether one and one is two. They take it as a fact.

I = 1 of course. — TheMadFool

What is one?

What is one? — Tzeentch

What is common to the sets {0}, {p}, {elephant}, {#}, {red}

2 threads + 2 threads = 4 threads, exactly? Not 1 river? — SimpleUser

We are speaking about mass not specific objects.
If we say 2 apples and 2 lemons we know it is not equal 4 because we are speaking of different concepts/objects. But, in the general (or mass) concept we have somehow “4” objects in the table because you count it. It is not infinite or zero.

Thanks very much for your reply. I would characterize your four sticks as "a very good argument that 2 + 2 = 4." But why do I call it "a very good argument"? Because it appeals to my intuition and only because it appeals to my intuition, I believe.

Regarding definitions, "2 is defined such that when added to another 2 it equals 4, and 4 is defined as the sum of two '2's' " would also be a good argument, but again because it appeals to my intuition and only because it appeals to my intuition.

"intuition isn't 'rejected' outright. . . . 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" is an answer that gets to the heart of my question (why academic philosophy seems to regard focus on intuition as a trivial and adolescent preoccupation). Does anyone here know of any formal papers that discuss that stumbling-block? But if that is the best response that academic philosophy has, I would reply:

1. problem-solving is not the only value that is relevant here; recognizing intuition's utility as a problem-solving tool can lead to less preoccupation with the problems, and more curiosity about intuition itself, and discovery of the value of introspecting (prompted by trying to understand our intuitions) in terms of psychological well-being and development

2. I think that introspecting, prompted by trying to understand our intuitions or by whatever prompts it, will help lead us to more and more correct intuitions – particularly, more and more correct moral intuitions.

[EDIT: I think that the observation "'2 + 2 = 4' cannot be proved, but rather rests on intuition," may be considered true but trivial, and may be considered trivial because you can't take it any further. But I would say that while you can't take THE OBSERVATION any further, you can improve your intuitions.]

There are two aspects to each of those statements. One is validity and the other soundness. In ordinary everyday reason, those are usually joined. They are explicitly separated only in mathematics, and sometimes in philosophy, but it always helps to treat them on their own, because they are naturally separable.

Validity is a matter of procedural correctness. Basically, you define the rules for your game and you play consistently. Playing with obscure variations is its own source of fallibility. For example, you claim that 3 + 1 = 4, that 2 + 1 = 3, and that 1 + 1 = 2, or more colloquially, you describe the successors of each number through addition. Then you describe the operation as being associative, i.e. (x + y) + z = x + (y + z). Then, you "prove" that 2 + 2 = 4, which is just another way of saying, that the game produces this statement as one of its many outputs. These statements are procedural by virtue of some system for formal deduction, which is essentially a qualifying description for a process that applies a set of rules. You can use this system to design steps to play the game. There are syntactic nuances that I am skipping, such as the use of terms, infix notation, etc., but those are not fundamental.

Soundness makes procedural validity applicable to life. Soundness depends on interpretation. With interpretations, you try to fit a square peg in a round hole, and claim that some part of your environment appears to match, approximately, probably, your procedure and its expressions. How does soundness occur. Through proper modelling, which is the inverse of interpretation. You take many different pegs with closely related shapes and you design a round hole to fit them all. Fingers, eggs, atoms, distance, etc. They all become numerico-analytic, i.e. quantifiable, by the simulation game you have designed.

Now, you may think that this does not apply to informal human reasoning, but merely to abstract mathematical formalization or mechanized computing. That is, I will contend, not true. Our mind constantly models its environment. Our brain structure implicitly plays games all the time, and tries its best to fit square pegs into round holes, as it navigates our circumstances. The driver of the modelling process are our survival instincts and our innate cognitive traits, themselves consequence from various homostatic and allostatic objectives that nature has, but it is still a matter of applying some approximately homomorphic rules to symbolic representations of the surrounding reality. Sometimes, our brain explicitly conceptualizes the implicit models, because we are gifted with self-aware cognition and then we create formal science. And finally, we transfer those rules to machines.

Now, what makes a statement true in practice? This question has two meanings, but they are related. Why is the procedural correctness ever interpretationally correspondent to fact? That is, why are the relationships between facts procedurally expressible? And second, why are facts at all relatable? The answer to both of these questions is - such is the nature of things. Representation by procedure and symbolism is possible, because nature is pervasively homomorphic, by which I mean that the different processes and features of objects in nature are similar enough, to be capable of consistent congruence under the proper design of their initial conditions. And processes follow consistent patterns and relationships over time and in space, because they have no other choice. The option of not doing so is an epistemic illusion. Whether those patterns and relationships remain consistent with our predictive description, is a matter that is subject to change, but they are themselves always appearing in the only way possible.

Note that 2 + 2 = 4, only if you are not in a cyclic group, which you will be if 2 is a walking distance on a circular path, with the perimeter being 3. And a square is rectangular in the Euclidean sense if the shortest distance paths between two of the adjacent vertices are traced by parallel lines, and that is only true if there is no gravity.

Thanks. That is very educational, and I certainly haven't digested it all yet. But at this point, let me just ask, does "those statements" in your first sentence refer to –

"'2 + 2 = 4' cannot be proved, but rather rests on intuition"

"'A square must be rectangular' cannot be proved, but rather rests on intuition"

– or to –

"2 + 2 = 4"

"A square must be rectangular"

– ?

Sorry, I was ambiguous. The latter. My aim was to clarify the meta-statements in my response, but I meant the basic ones in the text.

The meta-statements are both true and false. They are true in the sense that the soundness of the basic statements depends on the evaluation of fitness for purpose of axiomatic choices and rule selections that itself is not subject to formal proof. It is subject to implicit neurological representations that we have developed through history (personal or collective) of refinement, before they have even become explicated formally. Note that this is not magic. Our ability to model some parts or aspects of nature by encoding them in other parts and aspects is possible, thanks to nature's amorphity. A machine would be equally able, for better or worse, to emulate the refinement in the model selection process that our intuition captures, using trial and error, cost and reward criteria. (Where reward is chance of sustenance.)

The meta-statements are false in the sense that once the approximate abstract structure of some natural phenomena is captured formally (or much more generally, is neurologically encoded), you can prove (or mentally derive) the correctness of the basic statements without intuition.

I.e.:

'2 + 2 = 4' can be proved, but its interpretation and soundness rather rests on intuition
'A square must be rectangular' can be proved, but its interpretation and soundness rests on intuition

By the way, I made a mistake in my first post. I said ring in the last paragraph, whereas I should have said cyclic group. I don't practice my algebra.

But I would say that while you can't take THE OBSERVATION any further, you can improve your intuitions. — Acyutananda

What do you think improving one's intuitions would consist of? Aristotle placed intuition as the highest form of knowledge in his Nichomachean Ethics. He looked briefly at the question of whether intuition is innate or whether it is learned, and decided it was a combination of both.

In western society we generally consider intuition to be instinctual. It is the inherited aspect of knowledge. When you say intuition grasps the truth of "2+2=4", this would mean that we instinctually accept this as true. However, we still need to learn the meaning of the equation. We are taught it in school, so the instinctual aspect is the attitude that we have toward learning. We accept the teacher (authoritative figure) as the authority, we have a desire to learn, we see the usefulness in what is being taught, so it appeals to our intuitions.

How do you propose that it is possible to improve one's intuitions? Would this be a matter of moral training, to improve one's attitude toward authority? Or what do you think?

people don't believe you can build a society based on intuition. And thís is exactly the difference between science and philosophy. — TaySan

Thanks. I'm not sure if I understand. "people don't believe" sounds like "Nobody, or hardly anybody, believes." Yet doesn't "thís is exactly the difference between science and philosophy" imply that either scientists or philosophers do believe (you can build a society based on intuition)? Scientists take it as a fact that one and one is two. Is that believing in building a society based on intuition?

Thanks. I plan to get back to this, but I'll be tied up for some hours now. Specifically regarding how it is possible to improve one's intuitions, I think that the most basic answer is meditation, but I would like to say a little more.

I don't know! I never saw intuition like that. But for sure it isn't that black and white. That is the problem with language. It narrows down too much!

Thanks. I plan to get back to this, but I'll be tied up for some hours now.

For now, I just want to isolate one thing in your replies and use that as a springboard:

"axiomatic choices and rule selections. . . . is subject to implicit neurological representations"

Let's say that TheMadFool's "I I [2] + I I [2] = I I I I [4]" is the very best rational (as opposed to intuitive) proof there will ever be that 2 + 2 = 4 (at least as long as one is not in a cyclic group).

Though that is the very best rational proof there will ever be that 2 + 2 = 4, it seems to me that all that proof really achieves is to trigger in my brain a neurological event that neurologist Robert Burton calls a "feeling of knowing." (Sam Harris once referred briefly to this "feeling of knowing" – not specifically related to 2 + 2 = 4 – and did not provide any further details. But for me that is enough.)

To admit that that feeling of knowing is ultimately the only substantiation of 2 + 2 = 4 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, and in that sense the admission is trivial. But in the sense that we would want to know practically how to avoid/prevent occurrence of that same neurological event when we are contemplating 2 + 2 = 5; and in the sense that there might be tremendous benefits, including unexpected ones, if we could learn more, through brain scans (a kind of INDIRECT learning) about how those feelings of knowing occur in our brains in real time; or if, even better, we could introspect enough to have some DIRECT apprehension of how those feelings of knowing occur in our brains in real time; in those senses, the admission would not be trivial at all.

There are many things to consider. First, I think we agree that abstract reasoning is the game of token resolutions that substitute for directly established facts, such as 2 + 2 = 4, instead of those two apples and those two apples are enough to feed a single apple to each of those four people. Soundness depends on the proper selection of rules that govern reasoning in accordance to a set of matching observations. We need structure that amounts to what is essentially a generalized fact, but which can be inferred through a mechanical process. How do we establish a "generalized fact"? The design involves the extraction of unchanging patterns. We need to represent experience, according to some metric of practical fitness for purpose, according to some assignment of average utility, according to some perception of correspondence. As I said, approximately, probabilistically, homomorphically. There are two issues here. How to derive such representation and how to verify such representation. Although we know how to mechanically arrive at some simple models in particular cases, our technology to extract concepts from environment cues is still not quite there. Human beings rely on evolutionary, developmental, as well as ecosystemic features. That is, billions of years worth of evolutionary context, millennia of accrued culture, symbiotic group relationship, neural network that uses 100 billion nodes (neurons) and 100 to 1000 trillion connections (synapses), some of which specialized. So, emulating nature's job digitally, effectively even, is a pain. On a different note, we cannot design concise criteria of what makes a model good independent of the circumstances. Not involving a neural network is not really feasible, unless we are accessing some kind of apriori knowledge that does not depend on experience. Now, to confirm the value of our models, people can use evidence. We can also keep check through consensus and experimentation. Theories and axiomatic systems with infinite domains cannot be confirmed exhaustively, but because the general statements are usually applied in some restricted range (and are frequently not even expected to be valid outside of the scope of application), the intuition is mostly sound if it is confirmed through a comparatively small set of instances, whose coverage is judged according to various criteria, such as the perception of discontinuity and symmetry breaking in the physical structure of the objects, as well as the reasonable uniformity of the grid of instance placements, etc. And the rest of the range is assumed to reproduce the pattern. Something called inductive reasoning.

The second issue is about actual formalization. Our brain is capable of utilizing abstraction without explicit formalization, but the repetitive subconscious application of rules is not its forte. We want to design a system of inference that can be mechanized. Even if it will be handled manually, we want to separate the process of modelling and interpretation, from the derivation of theorems. There are a few comments here.First, 2 + 2 = 4 is rarely an axiom, but a theorem. You could make the argument that this is arbitrary, because it could be used nonetheless, but the successor relation follows a more basic correspondence to the repeating aggregation of unit values rooted in the physical and socio-economic structure of quantification. Second, when dealing with models that involve numeral constants, like this one, it is more useful to specify a schematic recurrence, and the successor recursive relation of numerals has a simpler, more elegant expression. Having infinite number of axioms, but specified in a single (or multiple) recursive schemata is called effectiveness. And, as a side note, you don't need numerals for arithmetic. You can simply specify the relations of various operations, and express 4 as "1 + 1 + 1 + 1" in every instance where it is needed. Or you can use the simpler unary system that encodes the result of "1 + 1 + 1 + 1" as "1111". Or you could use binary, which is more compact to axiomatize then other bases. And third, when formalizing science or mathematics and arithmetic, you want your minimal system of axioms. A property called independence. So, if you specify "3 + 1 = 4", you don't need to specify "2 + 2 = 4".

Did you notice that counting dashes or fingers is an act? It's not stating a definition as such, but rather showing how numbers are used.

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

Learning to count does not appeal to an intuition, it is learning to behave in a certain way.

Yes. Of course it is an act that we humans inside our knowledge call it “counting” just to put an order in our reality as you said towards how numbers are used. These concepts are very important for plenty of reasons.
Why a year is divided by 12 months? Order. Understand how numbers are used in our system.
Why the distance is divided by miles/kilometres? Again order, etc...

We see here how the act of counting leads us another group of

Meaning as use. — Banno

The use of put a stability in our reality.

Thanks. You have gotten me started thinking about some concepts that were new to me. I haven't understood everything. But I have understood correctly, haven't I, that among other things you have provided what may be some wonderful arguments to the effect that 2 + 2 = 4? They look like wonderful arguments, but arguments nonetheless. For an argument to convince me means, doesn't it, that it succeeds in triggering in my brain a pattern of synaptic firings that result in a subjective feeling of knowing (which is an intuition)?

Perhaps the best way to understand the role of the feeling of knowing is to look at this sentence of yours:

"the intuition is mostly sound if it is confirmed through a comparatively small set of instances . . ."

Couldn't we paraphrase this as "the feeling of knowing is mostly sound if it is confirmed through a comparatively small set of instances that result again in that feeling of knowing" – ?

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.

It's 11:30 pm where I am. I'll look for responses in the morning my time.

Learning to count does not appeal to an intuition, it is learning to behave in a certain way. — Banno

Right. But when I said "it appeals to my intuition, "it" refers to an argument that draws on my already-learned behavior of counting, doesn't it?

more curiosity about intuition itself, and discovery of the value of introspecting (prompted by trying to understand our intuitions) in terms of psychological well-being and development — Acyutananda

I have no issues with what you say and if I'd like to add anything it would be that:

1. Intuition deserves more of our attention because some of the greatest minds the world has seen have attributed their discoveries/inventions and whatnot to intuition and not mechanical, formulaic logical thinking.

2. Intuitions aren't always correct. Take for example the Monty Hall Problem. Per some sources, most people's intuitions tell them that there's no difference between sticking with one's first choice of door and switching doors after one of the doors have been opened but as it turns out that's wrong. This throws a spanner in the works for those who wish to examine intuition for its utility in cracking problems, easy and tough, because the success rate of intuition may be the same as that of guessing randomly. In other words intuition maybe just flukes.

3. 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, so to speak. However, when it's off the mark it can be very upsetting. It seems the knife cuts both ways.

more and more correct moral intuitions — Acyutananda

Perhaps this is related to your attempt to link intuition with "psychological well-being and development" and I agree to some extent with it. Back then, some 2,500 years ago or thereabouts, when people were just beginning to think about our sense of right and wrong logic was in its infancy and most of what they discovered about morality were/had to be intuitions rather than products of careful logical analysis. One exception though is Buddhism which comes off as a belief system that was founded on the bedrock of hedonism. I suppose the biggest hurdle in coming up with a sound moral theory was/is/will be our intuitions regarding good and bad as intuitions seem to bypass logic in most cases and that'll show when logic is brought to bear on our moral intuitions.

Did you notice that counting dashes or fingers is an act? It's not stating a definition as such, but rather showing how numbers are used.

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

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. A word's meaning is given by how it's used which, to my understanding, is Wittgenstein's attempt to draw an analogy between words and everyday objects.

Take, for instance, a book. While some may be of the opinion that a book has some sort of essence which can be captured by a definition, Wittgenstein claims that a book - what it is? (it's meaning) - changes with how it's used. For example if I hurl the book at a person, the book is a weapon; if I keep a cup of tea on it, it's a saucer; if I cover a bowl of hot soup with it, it's a lid; if I read it, it's a source of information; you get the picture. Each situation for the book in my example corresponds to a context for a word. Just as the book's meaning is a function of how it's used (in these different circumstances), a word's meaning is also a function of how it's used (in differing environments).

On the matter of numbers, it looks like Wittgenstein is N/A. The meaning of numbers is confined to mathematics i.e. for a number, say 2, there are no other contexts in which 2 has a meaning. In short, the meaning of 2 isn't a use thing.