Why is primacy of intuition rejected or considered trivial?
I have little formal background in Western philosophy, but I'm under the impression that in Western philosophy, 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 rejected, or considered true but trivial. If my impression is correct, I would like to know why such propositions are rejected, or considered true but trivial.
Comments (98)
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 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.
.Quoting 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.
Sticks, fingers, luxury yachts. They don't answer the question of what is one.
I = 1 of course.Quoting javi2541997
:up:
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
2 threads + 2 threads = 4 threads, exactly? Not 1 river?
What is one?
What is common to the sets {0}, {p}, {elephant}, {#}, {red}
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.
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?
Quoting 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.
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".
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 Quoting 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.
Quoting 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?
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.
Quoting 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.
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.
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).
Quoting TheMadFool
This is not what I meant. Let me try to summarize my argument. To see what I meant, please see my 5. below.
Quoting 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, Quoting 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.
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!
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.
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?
I don't see the relevance...
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.
We can.
An example?
Quoting 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.
Quoting 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:Quoting 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.
So then 2 + 2 can equal 22
May I?
"Would you care for another glass of 'Two Barrels'*?"
*it's a brand of Whiskey.
"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.
That's a play on words, both meanings are using "two" as a number.
This is confusing syntax and semantics. Strings are infinitely dimensional space, we are talking about the concept of integers, which are ordered set.
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.
:rofl:
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.
Quoting simeonz
So my attempt was successful(?).
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.
Great but "Barrels" spoils the show in a manner of speaking.
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.
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.
:up:
I am not sure how you mean it. Human beings, I believe, are capable of classification according to features and of mental homomorphic representations without explicit involvement of applications, just by physical assessment. There is obviously always some action involved, because observation or measurement are usually tied in to some action, but the concept is not always action-oriented. It depends on the concept, and how we mean it.
For example, integers and natural numbers, as most discrete mathematics, are fundamentally tied in with counting. They are procedural by the very essence of their design, because without discrete generating processes, discretisation has no subject matter. There are other ideas involved, such as the equivalence of objects, but counting is the central idea. Real numbers are also inspired by a process, i.e. measurement, a.k.a. geometry. Analysis explains how we bridge the gap between counting and geometry through sequential approximations. In this sense, the intuition for real numbers is also procedurally inspired, but the subject matter of physical relations that we conceptualize through analysis and analytic geometry is not the actual application of a process, but the features exposed through the application of a process. Obviously, geometric intuition, as the name implies, is derived from the need for measurement, and obviously measurement is how we establish the relations in question, but unlike natural numbers where the procedure is the actual concept, geometry describes characteristics independently of the process involved in their determination. Still, I admit that you could argue that at the very fundament of physics, the determination of any geometric feature is connected to the time it takes for bosons to reach from one point to another, so you could argue that all manners of measurement processes are actually fundamentally related. But I would consider this argument belongs to a more fundamental discussion that deals with the static vs dynamic as a distinction in nature.
On the other hand, there are several other ways in which you could mean that actions are relevant here. First, counting specifically (for small quantities) may have preceded conceptualization historically, which may have appeared even before language did. So action may have preceded abstraction, because conceptualization, which apparently was sprung as a faculty linguistically, was not available and rote learned behavior was pertinent. Second, to establish the practicality of concepts, we evaluate their utility, which is usually ascribed to process application, i.e. uses as you put it. Lastly, inference itself is a procedure and usually well exemplified by correspondent real world process.
Edit.
I should have also stated that weighing gravitational mass and measuring time may have also had influence on the development of analysis. Thus, the study may have unified multiple procedurally unrelated subjects. Weighing and geometric measurement could have been seen as a totally separate aspects. However, now we know that inertial and gravitational mass coincide, and we can describe inertial mass, as well as other properties, such as energy, momentum, etc, through the trajectories of motion, and thus geometry. Frequency, albeit a general concept itself, encompassing all sorts of periodic phenomena, usually manifests as mechanical and field waves, either related to momentum and thus to the geometry of motion, or directly to geometric distances between sequential wave peaks and troughs. The measurement of distance itself, originally considered a subject of isometric application of a fixed template object, today is expressed as the rate of travel of electromagnetic signal in a fixed amount of time. Time in turn is measured relative to the period of electromagnetic wave that is emitted by specific energy transition in the electron configuration of an atom. What I mean by all this, is that currently, considering all our knowledge of physics, we can tie most applications of real numbers to the process of electromagnetic radiation.
I disagree. Each concept has a place in a language game. But then, for most of you post I agree with .
@TheMadFool, incidentally, Fool's question was about "2", not 2, so I don't see that emancipate's syntactic answer was improper, nor @Isaac' plonk, but moreover @frank's lawnmower is a semantic example.
Quoting frank
That's not what was suggested. Rather, that in this sort of analysis we would do better to look to use rather than to meaning. As your lawnmower example shows.
And I will make the point again that I do not see any relevance to this digression.
No, because if the whiskey were called 'Three Barrels' and I asked for 'Two Barrels', I would not be given two thirds of a whiskey, I'd just be given the wrong brand. 'Two' in that sentence is not being used to count, it's being used to indicate the type of whiskey wanted (not that it ever would, Two Barrels is awful).
But yes, this is a digression. As you were.
Not to put too fine a point on it, but branding doesn't obliterate the original meanings of the words involved. "Dos Equis", a so-so beer, has a name that should be understood to refer to two x's. Number used as name.
:rofl:
The syntactical or neurological processing is undeniably an action. Symbolic inference using a deduction system is an action. Thinking is an action. In that sense, mathematics is alive, not stationary. And formal models are sound with respect to the matching between those evaluations made using rules and the facts of the real world. I got the impression that you consider the subject matter of a model, i.e. its interpretation, to be some action in itself. In other words, I thought that you might suggest that before we formalize stationary physical relations around us, we first discover them in terms of applications, uses, and activities. That without some active involvement on our part, it is insufficient to simply observe physical features in order to derive conceptualization. I was trying to clear out in what precisely sense you meant your remark.
Quoting Banno
When TheMadFool quoted the numeral, I assume that they meant to emphasize the syntactical nature of the symbolic constant. Not to prompt interpretations that do not conform to the algebraic requirements posed by the axioms of arithmetic.
If you subscribe to total moral and intellectual relativism, then of course there is no such thing as a correct intuition.
I don't see how it could be otherwise.
Perhaps Wittgenstein is the go to person here. Moral intuitions, their variety and seeming incompatibility, is probably due to the fluidity of the concepts good and bad as they participate in distinct language games. Moral relativism assumes, contrary to Wittgensteinian thought, that good and bad refer to the same things, hence the relativism; without that assumption, moral relativism would break up into separate entities which have nothing to do with each other. Right?
Do you mean "good and bad refer to the same thing [singular]," which would mean "good and bad refer to good-bad," or do you really mean "thingS," in which case the things would be something apart from good and bad? If you mean the latter, what would be examples of the things?
Quoting simeonz
The syntactical or neurological processing of what? An action on someone's part would seem to require some fiat on their part... as if my feeling a surface somehow required my consent.
What I mean to convey is that it's possible that "good" and "bad" may refer to different concepts in different cultures i.e. morality may not be amenable to generalization across cultures. This is Wittgensteinian in character as you might've figured out by now.
Why bring Wittgenstein in at this juncture?
Well, moral relativism, to my knowledge, claims that moral values vary with culture with all of them being as equally right. However, for that to be true, the notion of morality has to be universal in scope i.e. every culture must mean the same thing when they use the words, "morality", "good", and "bad". If not then moral relativism doesn't make sense for then different cultures would be talking about different things when they use these words.
So you are saying that –
moral values vary with culture
– and –
every culture must mean the same thing when they use the words, "morality" . . .
– can both be true at the same time (and must be in order for W to be correct). In order for both to be true at the same time, your "moral values" must mean something different from your "'morality', 'good', and 'bad'."
Does your "'morality', 'good', and 'bad'" refer to CONCEPTS, while "moral values" refers to SPECIFIC ISSUES?
Would you say, for instance, "In Mexico they consider bull-fighting good and abortion bad, and in NYC they consider abortion good and bull-fighting bad, but in both cultures, when people hear 'bad' their minds become clouded and there's a yucky feeling in the stomach, and when people hear 'good' their minds become expanded and radiant" – ?
I guess I now understand what you're saying, but – perhaps changing the subject – wouldn't you say that in order for the terms "good" and "bad" to be most useful, to be truly normative (using your examples), "good" in Iran would have be associated not only with the Quran, but also with a positive quasi-emotional feeling, and "good" in America would have be associated not only with equality, but also with a positive quasi-emotional feeling?
We would all converge to a point (a set of ideas) but the cost of that is we'd be losing out on the richness of human thought.
We can all agree on word definitions, for the sake of communication, without agreeing on anything else. People of all cultures can agree that "good" should be associated with a positive quasi-emotional feeling, even if in some people that feeling is elicited when they hear "marry whomever you love," and in others when they hear "throw homosexuals off rooftops." I wouldn't see any value in lack of agreement that "good" should be associated with a positive quasi-emotional feeling, and thus people talking past each other.
I.e., I wouldn't call people talking past each other, due to lack of agreement about defs., "richness," because to me "richness" has a positive meaning.
Aah! You mean to say that "positive feelings" are not a good yardstick for morality. I agree for the reasons that you put forth. Americans may feel good when they see homosexuals marrying each other but an Iranian may experience the exact same "positive" emotions when they see homosexuals taken to the gallows.
Yet, there seems to be something generic about morality in re "positive feelings" - we feel good, barring some exceptions, for the same things - a partner, a family, friends, food, health, helping others, shelter, clothing, amenities, etc. Should we ignore the many instances where "positive feelings" are aligned to our ideas of morality and focus only on the cases where the two don't concur?
I mean that they ARE a good yardstick for the meaning of the word "morality," but not a good yardstick for which actions are genuinely moral. I think that some actions are genuinely moral and some are genuinely immoral – in other words, I believe that an objective morality exists. (This discussion all goes back to my saying "Intuitions can become more and more correct. . . . I think that we all, to different degrees, possess deep in our minds a capacity for more and more accurate feelings of knowing," and your feeling that that statement of mine did not jibe with something else I had said.)
Let me illustrate. Suppose there are a dozen Americans and a dozen Iranians who agree with me that "good" should be applied to actions that elicit in the speaker a positive feeling – not an emotion exactly, but definitely a feeling, not part of any rational process, a unique feeling often called a moral intuition – and "bad" should refer to actions that elicit in the speaker a unique negative feeling. If the action is one's own, the mechanism that causes the positive moral intuition or the negative moral intuition to arise is known as the person's conscience. So all these people will completely understand what the others mean when they use the words "good" and "bad".
Nevertheless, those particular dozen Americans will say, "I did something good today. I feel good about it, moral about it. I helped a couple in some fundamentalist country have a same-sex wedding." And those particular dozen Iranians will say, "I did something good today. I feel good about it, moral about it. I helped get a couple hanged for having a same-sex wedding."
Yet I would not say that all those people's moral intuitions are equally valid (moral relativism).
Let's find the most uncontroversial example we can. If someone experiences the negative feeling, the "This is bad" moral intuition, when they see someone torturing puppies for fun, I think that is an objectively correct moral intuition. If the torturer and his friends have a different moral intuition about it, I think their intuition is simply incorrect, and should not be framed as "their truth" or "all right from their point of view."
And I think that any person's moral intuitions can become more and more correct. When asked me how they could become more correct, I replied, "I think that the most basic answer is meditation, but I would like to say a little more." Then other participants sidetracked me from that discussion.
To see this, you have to try to conceive of something as being irreducible, which is what an intuition is supposed to be, a non discursive disclosure, a "knowing" that issues straight out of the belly of the world, unmediated by anything else. If something like this actually existed, it would be monumentally important, as if God had written it on tablets, but with out the God or other weird metaphysics. As if Being "spoke" its nature.
Of course, logic and math are intuitive, you might say, but is it true that these are simple, irreducible things? Just because thought cannot get "behind" logic, because doing so would require logic, doesn't mean logic is IT. It is not accessible, but only shown, but, as Wittgenstein tells us, even to talk like this is nonsense. A true expression of intuition cannot be made sense of at all, for the language one is using to express it cannot stand apart from itself. even if it could, a third perspective would then arise to validate this, a third medium that is NOT logic, and this is not conceivable.
Thanks. I need a few clarifications, but let me start with the easy part, which is also a necessary part. Do you mean that correct intuitions (which I believe exist), would be monumentally important, as if God had written them on tablets?
I believe that correct intuitions exist and are monumentally important. But I don't see why for every correct intuition that exists (there is a maximum of one correct intuition per issue), I should consider all the many incorrect intuitions that might exist on the same issue to be monumentally important.
I would not call any intuition, correct or incorrect, important or not, irreducible. My intuitions are by definition feelings (of a certain sort) whose origin is presently unknown to me. An intuition comes out of my unconscious in some way that I cannot understand. But I think it must originate in some way that I cannot presently understand – must be reducible to something that I cannot presently understand – but that I may (or may not) later be able to understand.
If I understand correctly, Wittgenstein's main works were the Tractatus and Philosophical Investigations. Can anyone tell me what writing of his in those books or elsewhere best addresses my concerns?
“The concern with grammatical propositions was central to Wittgenstein's philosophy of mathematics because he wanted to show that the 'inexorability' of mathematics does not consist in certain knowledge of mathematical truths, but rather in the fact that mathematical propositions are grammatical. The certainty of'2 + 2 = 4' consists in the fact that we do not use it as a description but as a rule.” Ray Monk
But it is the very notion of intuition itself that is at issue. Does the concept make sense at all, apart from the casual talk about premonitions and feelings? If you try to give an example of an intuition, can it be defended as truly irreducible? How about my favorite intuition, causality? Try to imagine a spontaneous effect, ex nihilo. Can't be done; I mean knock down, drag out, impossible, a causeless event. Why? Just the way it is. Then there is logic, geometry, math and pretty much that's it. These amount to the same thing, it has been argued, but note, there is no answer to the question, why? Why does the leg opposite the largest angle of a triangle have to be the longest leg?
This is about as close to a genuine intuition that I can imagine. I don't deny that such things are intuitively apprehended. It gets difficult, here, frankly. 1) Is Kant right, and our intuitions are just representations? This would place it in Kant's world, where we are "shown" things through intuition, but intuition is not that which is shown. Its foundation is unknowable. Therefore, intuitions are constructs, and therefore contingent. 2) Enter Derrida's world. I don't know this that well, but as I see it, all thought is part of a web of contingency, no one idea stands alone. I say modus ponens, but these term get there meanings, their sense, from their play against what they are not. Up makes no sense unless down is there to be posited. There IS no stand alone proposition, for all ideas are like this. Thus an intuitively apprehended truth is really an embedded truth, for each part of the utterance is utterly senseless in and of itself. This makes all truth contingent.
"But it is the very notion of intuition itself that is at issue. Does the concept make sense at all . . ."
Thanks. Your later lines make it clear that you do not deny the existence of intuitions. Since you think they exist, you must think that they make sense IN SOME WAY – at least as "premonitions and feelings" that don't necessarily amount to anything, as you say.
We seem to agree that intuitions are feelings that exist. We differ in that I think some intuitions are correct, that is, objectively true, and that such intuitions don't just occur randomly, at least not always randomly.
"If you try to give an example of an intuition, can it be defended as truly irreducible?"
I, at least, do not try to defend it as irreducible. As I said, "I think it must originate in some way that I cannot presently understand – must be reducible to something that I cannot presently understand . . ."
Just to show that this thinking is not incoherent, let me give an example that a Christian might give. I think a Christian might say, "Jesus's moral intuitions, the feelings he would immediately have when any moral issue was brought to him, were invariably correct. There is a reason why they were invariably correct, and the reason is that God intentionally created/fathered Jesus so as to have correct moral intutions. Jesus's moral intuitions were not irreducible – they were reducible at least to an act of creation/fathering by God."
I am not a Christian, but I think that while people may start out with all kinds of bizarre moral intuitions, we can all develop ourselves so that our moral intuitions, the feelings we immediately have when facing any moral issue, are increasingly correct.
This much of your post seems to be in almost complete agreement with me. The only difference between us seems to be your "as close . . . that I can imagine." Why not just say "This is a genuine/correct intuition," as I do?
Could not your "there is no answer. . . . apprehended" be paraphrased “The correctness of this geometric principle/proposition cannot ultimately be proved by any discursive argument. Its correctness ultimately rests on intuition, Such intuitions are intuitions that almost everyone has, and they are correct intuitions" – ?
"we are 'shown' things through intuition, but intuition is not that which is shown."
Can you refer me to where Kant says this? Anyway, I agree.
"[Intuition's] foundation is unknowable."
I gave my opinion earlier: "An intuition comes out of my unconscious in some way that I cannot understand. But I think it must originate in some way that I cannot presently understand – must be reducible to something that I cannot presently understand – but that I may (or may not) later be able to understand."
I wouldn't give up on eventually understanding.
"Therefore, intuitions are constructs, and therefore contingent."
Are you still representing Kant here? I don’t see why this should necessarily follow from "[Intuition's] foundation is unknowable." Let’s take my Jesus example above (which I don’t believe in, but which I think is a coherent story – not empirically true, but not a story that violates logic). Jesus may not know where his intuitions came from, and may never know (in which case they are unknowable to him or perhaps to any human being); nevertheless, God put those intuitions in him; so they are not just a construct, and not contingent.
"[Intuition's] foundation is unknowable. Therefore, intuitions are constructs, and therefore contingent."
Would this be your answer to my "Why not just say ‘This is a genuine/correct intuition,’ as I do?"
I don’t see how any uncertainty in knowing the foundation of truth necessarily makes truth contingent. Again, my Jesus example. The Jesus in my story may not know where his intuitions came from, and may never know (in which case they are unknowable to him or perhaps to any human being); nevertheless, God put those intuitions in him; so they are not just a construct, and not contingent.
“2) Enter Derrida's world. . . . This makes all truth contingent.
Can't we distinguish between truth and knowing truth? Derrida must have had some answer to this, but was it a convincing answer?
If I understand correctly, Wittgenstein's main works were the Tractatus and Philosophical Investigations. Can you tell me what writing of his in those books or elsewhere best addresses my concerns?
A computer can produce mathematical proofs through syntactical processing and it does not need any fiat at all. I meant action in the sense we talk about in mechanics - some process. But the point was that only the representation needs to be manipulated in accordance to the logic and algebraic rules of the abstraction, in syntactic or neurological terms, and that we don't need to test soundness through its application and involvement in an actual situation. We can derive and confirm soundness through observation that has no relation to any sense of utility. We don't need to be practically involved. We can be observers from a remote perspective. For example, by watching the stars in outer space.
The conceptualization of mathematical abstractions from experience may appear similar to any other form of cognition, but in specific detail, the involuntary subconscious impulses to sensory information become subject to contemplated analytical effort by the observer. Yes, it is still basically neural networks firing, signaling through synapses, cascading neuron activations, but the part of the subject that we would call reflex gives way to the part that we would call intent. At least in contemporary mathematics and sciences. Some primitive mathematical practices may have appeared so early, that it may have been a reflex for a wile.
And I do agree that fundamentals in logic, mathematics and sciences may be situated cognition, i.e. proper behavioral alignment with the environment. But this applies to few ideas - logic, induction, probability, etc. The rest, I conjecture, can and are derived by model extraction without first hand practical experience.
But how would Wittgenstein handle international relations, culture exchange, global politics? In the end, aren't we all one society with internal boundaries?
That is, morality may have local meaning, but that does not preclude it from being integrated into a global system of meanings that arbitrates and rejects.
As I have articulated in another post, intuitions are discovered contingently, but truth is only one possibility behind their conception. There are wrong intuitions. So, when we deal with mathematical abstractions, we usually use a small number of primitive intuitions for which we have consensus and large volume of experience and derive the remaining features of our models in accordance to those intuitions, without proliferating science with myriad of novel intuitions. For example, real numbers as complete ordered set, or completion of the rational numbers, are not exactly prima-facie concept and their soundness is doubted, because we are trying to justify it empirically. As long as we believe in inductive empirical reproducibility, objectivity of sensory experience, rationality of nature, statistical discovery of utility, we can assess the soundness and uses of various concepts. But I agree that those basic intuitions we trust blindly and without justification... (And I should say, not necessarily to a good conclusion.)
Arguing is too strongly put. I am expressing opinion. Namely, that only the very fundamentals of mathematics are situated cognition, but most of it relies on modelling through observation and contemplation.
From the same link
I am left with the impression that this is utilitarian model of knowledge. I propose that only the basis of mathematics, such as predicate logic, induction, probability, are derived by use, and the rest can be extracted by vain observation of nature or articulated on top of other abstractions, without seeking actual non-epistemic benefit from those models. I believe that the work of Riemann in topology originally lacked applications and found most of its practical uses later.
As to Kant, his transcendental idealism is all about representation. When an idea is put in place, and its logic on tact, this structures our world, but then there is the problem of noumena. This impossible "otherness" of all of our thoughts and experience applies to sensible intuitions as well as the transcendental ego of apperception. I would have to go over this again, read the Deduction and, well, the whole thing, but Kant's analysis takes us to the point where explanations run and even sensible talk runs out, which is why he was so reluctant to talk about noumena. It is unspeakable.
Wittgenstein made it a point to disarm knowledge of its power to penetrate the world absolutely. His argument, among others, was this: for a concept to be sensible, its opposite has to be conceivable. So to talk about something beyond thought, beyond logic, is not even conceivable. When we are faced with the question, what is logic? we really don't have even a possible response, for to have this would require something that is not logical to describe it, lest the question is begged. So while we are irrevocably IN logic, we cannot say what it is.
The "what" of what is shown is impossible to understand. One can only use logic, and observe its structure "through logic itself". All explanations are, after all, propositional.
Take a look at Rorty. Is logic really foundational? Or is it something else? How about pragmatics? I was once an infant child, a world of "blooming and buzzing". How did I come to know the world? Language was modeled, its sounds filled the air and associated with objects, experiences. Thus, IF I say a word in context C, THEN I get lots of smiles and encouragement. This conditional is foundational to language and the experience of the world. The logic of the conditional issues from this pragmatic engagement that is a basic condition to survival and reproduction, an evolutionist would argue, and the conditional is really a pragmatic function of problem solving. The scientific method is this, not a formal condition of ultimate reality, and it is science that ruled the child's acquisition of language skills. The brain evolved to be an instrument of conditional resoning because interface with the environment was essentially conditional.
Time is foundational for logic is executed in time, so it has a beginning, a middle and an end. Dewey called this end consummatory. Logic is essentially a pragmatic, a useful instrument.
Do I buy this? Yes and no. Always has to be kept in mind to conceive of anything, it is done through logic, so a pragmatic theory of logic, presupposes logic in its theorizing, for this talk about pragmatics is logical talk. This makes logic antecedent to pragmatism, doesn't it? But then, the argument itself that places logic as antecedent is itself cast in logic. So where does this lead us?
It leaves us with best guess when it comes to anything.
Is knowledge an intuition? I know my cup has coffee without looking. Memory, of course. But memory simply "comes" to you. And how do you know you can rely on memory? Memory tells us about events in the past with a great deal of accuracy. So we have this system which is essentially pragmatic: memory works, it's consistent, and why is consistency preferable? Because it works.
It is a strange business to be like Dostoyevsky's Underground Man and intentionally make great pains to spit in the face of reason and logic (using reason to do so, of course), but frankly, I lean toward your position when all is said and done, for it is not reason the underground man objected to; not really. Reason has no content. It is an empty vessel that neither denies God nor affirms foundational meaning. What the objection is really about is the presumptions of knowing that true conclusions follow from true premises, and here is what is true....Such confidence is ridiculous. A great book: Shestov's All Things Are Possible, in which he shows us how this confidence is groundless. We "know" nothing of the foundation of all things, yet meaning, not Frege's' "sense" of what ideas have, but affect, moods, joys and tragedies, these are the things at the center of the human condition, not clarity is what is true.
The significance of our intuitions lies not with reason, the empty vessel, but with the meaning it carries: the Good! And the Bad! What does it mean that I "enjoy" this danish with coffee? Or love Van Gogh and Ravel? What does it mean to be happy? Or to suffer terribly? These are the things humanity is about, and they simply are "there" and are complete mysteries. Intuitions.