Notes on A Priori Knowledge

[Pythagorean]

Ever Since Plato Philosophers

Have been impressed by the fact that there are universal propositions which we can apparently know to be true with absolute certainty, though we could not possibly observe all, or even a large proportion of the instances to which they apply. Such propositions, called "necessary propositions" by philosophers, are especially to be found in mathematics, the science which establishes its theorems with such infallible certainty that the expression "mathematical certainty" has become part of the everyday idiom. In Plato's dialogue Meno, a slave boy is led by Socrates to "see" that a square whose side is a diagonal of a given square has exactly twice the area of the latter (the proposition in question is a special case of the "Pythagorean Theorum"). How can we know that it is so in every possible case of a square with inscribed diagonals? How can we know that not only the triangles we have drawn on the blackboard in order to "verify" by measurement that the internal angles add up to 180 degrees have an angle sum of 180 degrees but that all conceivable triangles have it? Philosophers who hold that experience is the only source of human knowledge ("empiricism") may, like John Stuart Mill, say that our conviction is nothing but a habit of association, built up by repeated observation that one property is conjoined with another, but a philosopher who holds such geometrical knowledge to be independent of experience, a priori ("rationalism"), retorts: If such were the case, why are we not convinced to an equal degree that all crows are black, or that all bodies have weight, or that the ground gets wet whenever it rains? It seems that we can conceive of exceptions to the latter propositions (even if we find it hard to believe that there ever will be any), in a way in which we cannot conceive of exceptions to the former propositions. If we found a square whose side is the diagonal of another foursided figure, yet whose area was not double that of the latter, we would conclude that the latter figure is not exactly square: indeed, we would abandon any previously entertained beliefs that were relevant except the belief in the absolute validity of the theorum.

At any rate, this is the way Plato, Kant, Leibniz, and many other philosophers felt and feel about mathematical knowledge. In describing it as a priori knowledge (following Kant's terminology), philosophers refer to its apparent independence of experience: we claim to know that two pebbles and two pebbles make four pebbles even on Mars, before having verified this by actual counting after a strenuous trip in a rocketship. Perhaps, we also believe that if there are crows on Mars they are black, yet as we would admit that this is only probable on the basis of past experience: it is conveivable that on Mars or on some other planet there should be animals which are exactly like the animals we usually call "crows" except that their feathers are, say, red. In this sense our knowledge of the proposition that all crows are black is said to be empirical, and traditionally empirical knowledge has been said to lack that certainty which attaches to a priori knowledge.

It is important to understand in exactly what sense a priori knowledge is "independent of experience." No philosopher has ever denied that a child has to learn that two and two make four by learning to count, and that the latter process involves contact with concrete objects. But this only means that without sense-experience one cannot acquire the concepts of number, in this case the concepts "two" and "four," that is, a child who has never learned to count, to associate different numerals with distinguishable objects, will not even understand what "two and two make four" means. What the philosophers who believe in a priori knowledge assert is only that once the concepts have been acquired, the proposition can be "seen" to be true by just thinking about it (by "the mere operation of thought," in Hume's phrase). Consider the statement "for any objects A, B, C: if A is bigger than B and B is bigger than C, then A is bigger than C." Obviously, a being without sense of sight and sense of touch would have no concept of the relation designated by the word "bigger," hence such a being would not even understand that statement. But in saying that its truth is independent of experience, philosophers only mean that anyone who understands it will see that it is necessarily true, that it could not possibly be refuted by any observations at any time or place.

The philosopher not only makes the distinction between a priori knowledge and empirical knowledge by reflecting on the appropriate methods of justifying our beliefs; he goes on to ask how a priori knowledge is possible. Plato apparently was so perplexed by the fact that we can know universal propositions independently of experience that he had to invent a myth in order to account for it: the soul remembers visions it has enjoyed in a former disembodied life. Other philosophers, less poetical than Plato, tried to account for it in terms of a distinction between two kinds of entities, a distinction that played a vital role in Plato's philosophy: universals (Plato called them "forms"), and particulars. When we look at the blackboard we see particular triangles, but when we prove the Euclidean theorem about triangles we think of the universal triangularity, i.e., that which all the particular triangles have in common and by virtue of which they are all triangles. Every particular triangle has a particular size, for example, but when we classify it as a triangle we abstract from this particular feature and focus attention of a property which it shares with similar figures; it is this common property which philosophers call a universal. Again, we can see particular cubical objects at different places at the same time, or at the same place at different times, but when we think about the nature of a cube (as when we say to ourselves "every cube must have twleve edges") we think, in the terminology of those philosophers, about a universal that is identically present in all visible and tangible cubes. Whenever we classify a particular thing or event--in short, a "particular"--as being of such and such a kind, we consider it as an instance of some universal, or set of universals tied together by a single name, like "cow," "man," "table," "rain," "thunder." According to Locke's doctrine in the Essay Concerning Human Understanding and Russell's in The Problems of Philosophy, we can be certain that every particular which is an instance of universal A is also an instance of universal B, though we can never survey all past, present and future instances of these universals, if we "see" with our intellectual eye a certain relation between A and B, a relation which is sometimes called "necessary connection," sometimes "entailment." If we can see that squareness entails equilateralness, and that being a cube entails having twelve edges, then we can be sure in advance of sense-experience (think of the original meaning of "a priori" in Latin: before!) that there are no squares that are not equilateral, nor cubes that do not have twelve edges.

[Aedes]

I've always had difficulty accepting the proposition that "1 + 1 = 2" is truly a universal. The reason is that it's not really a verifiable proposition -- it's an equation whose constituents are already defined relative to each other. The information content in '1 + 1' is absolutely identical in '2', just as they are absolutely identical to the cubic root of 8. So to call '1 + 1 = 2' a proposition or even a relationship is a redundancy. To say 2 always implies 1 + 1. The distinction in terminology is useful only because '1' constitutes something different independently.

By extension this is true of the pythagorean theorem, because in an equation you can always break down the constituents to expressions of equivalency. The terms define themselves, so any equation is ultimately a circular statment.

I think Russell, Frege, and others tried to demonstrate that all of mathematics were reducible to pure logic. But this was rebuffed by Godel's incompleteness theorem, which shows that not all of mathematics is logically demonstrable.

[Pythagorean]

Originally Posted by Aedes

I've always had difficulty accepting the proposition that "1 + 1 = 2" is truly a universal. The reason is that it's not really a verifiable proposition -- it's an equation whose constituents are already defined relative to each other. The information content in '1 + 1' is absolutely identical in '2', just as they are absolutely identical to the cubic root of 8. So to call '1 + 1 = 2' a proposition or even a relationship is a redundancy. To say 2 always implies 1 + 1. The distinction in terminology is useful only because '1' constitutes something different independently.

By extension this is true of the pythagorean theorem, because in an equation you can always break down the constituents to expressions of equivalency. The terms define themselves, so any equation is ultimately a circular statment.

I think Russell, Frege, and others tried to demonstrate that all of mathematics were reducible to pure logic. But this was rebuffed by Godel's [url=http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems[/url] incompleteness theorem, which shows that not all of mathematics is logically demonstrable.

Aedes, don't you think that there must be something within space and time or within the nature of things that we must intuit a priori in order for it to be possible for us to construct positive propositions which would yield new knowledge? Something that our propositions conform to besides other propositions?

It seems to me that if we rely upon propositions to validate still other propostions in the manner of empiricism(?), then we do fall into the trap of scepticism aimed at the method of induction which is reliant upon a supposed consistency of nature to supply a regularity of particular appearances. But such a regularity of particular appearances is doubtful. Since nature itself is imperfect at the level of her empirical sources, we can't be certain that particular phenomena will appear; appearances will therefore remain only probabilistic or indeterminate in nature. This is, I would claim, because nature in her particular effects does not inherently conform to reason but is rather shady and imperfect in an external sense. Also, there is no knowledge to be gained from the investigations of single cases; therefore knowledge must be prior to sense experience.

So I would say that such organizing principles are not to be found at the external points of nature, and that only uncertainty is ultimately found there. The plan to organize nature does not come from nature, of course, but it is only through the sciences that nature is partially conformed to the will of man or partially organized by man. The grounds for these positive constructions on the part of man i.e. the 'plan' to organize nature, are to be found in our needs and desires, and ultimately in the human mind's reasoning capacity.

So, I disagree with what you seem to imply because it seems obvious to me that constructive positive knowledge is possible and such knowledge cannot be through and through a tautology. Constructive knowledge is a result of man's ability to 'put two and two together', a capacity which the brutes do not have and I see no other source for this capacity but that it exists a priori.

--

[Aedes]

Originally Posted by Pythagorean

Aedes, don't you think that there must be something within space and time or within the nature of things that we must intuit a priori in order for it to be possible for us to construct positive propositions which would yield new knowledge? Something that our propositions conform to besides other propositions?

Yes, but intuiting something a priori does not prove or depend on its universality -- it only presupposes it.

So, I disagree with what you seem to imply because it seems obvious to me that constructive positive knowledge is possible and such knowledge cannot be through and through a tautology. Constructive knowledge is a result of man's ability to 'put two and two together', a capacity which the brutes do not have and I see no other source for this capacity but that it exists a priori.

But insofar as 2 is synonymous with 1+1, 1.9+0.1, 1.99+0.01, 1.999+0.0001, etc, all numbers and their relationships are a tautology. The reason that there cannot be another realm in which 1+1=5 is simply that that would contradict the way we define those terms. The universality germinates from our conventions, and it's our collective agreement on the meaning of concepts and words that allows us to assume their universality. But short of pure objectivity and true omniscience, it's not logically necessary for us to conclude or even presume that something known a priori contains universal truth.

[Pythagorean]

Originally Posted by Aedes

Yes, but intuiting something a priori does not prove or depend on its universality -- it only presupposes it.

I do agree, and I think it's quite interesting to think about. Simply because there must be something that we 'intuit' and upon which depends the advent of our positive knowledge does not necessarily entail the existence of universal, necessary, forms.

The philosophical and metaphysical 'systems' of Descartes, Spinoza and Leibniz, as far as I can see, do not posit the existence of universal forms. But what these 'rationalist' thinkers have proposed is that the primary criterion on which knowledge rests or is justified is intellectual and deductive. It emphasizes the role of reason within us, as patterned after mathematics, in order to account for any science.

What I would propose is that the entirety of nature, whatever it is, is intertwined and therefore universals cannot be said to enjoy an independent status but must be unified, interconnected and somehow available to us through reason or as the reasoning process in action.

But I believe that to say that universals exist is a way of saying that the possibility of reason places or implies a limit upon the available types of natural appearances or phenomena; that appearances can be systematically described and positive knowledge can be actual because of this inherent limitation on nature's available classes or 'types', which is not to say that any single supposed universal posesses a priviledged ontology in its own right. True omniscience is therefore, as Leibniz has hinted at, a distinct possibility. Omniscience being only subjectively obtained and defined as man's posessing a completely articulated and fully known position within a given limited, and probably a scientifically generated, environment.


I propose such a scheme because I think that if nature is said to be infinitely open regarding the production of natural 'classes' or types, then reason cannot, will not be grounded, cannot be said to posess real 'traction' at all and the ultimate justification for our knowing capacity is therefore illusory. Therefore, intelligibility or knowability must be limited not at the level of empirical nature (which is infinite and I believe merely probabalistic) but at the level of which our reason operates.

So that there is what one might call a 'singularity', a point of absolute 'substance' or meeting point between the knowing mind and its object, which is intelligible and not empirical, and which is the absolute upper limit at which reason can operate.

But insofar as 2 is synonymous with 1+1, 1.9+0.1, 1.99+0.01, 1.999+0.0001, etc, all numbers and their relationships are a tautology. The reason that there cannot be another realm in which 1+1=5 is simply that that would contradict the way we define those terms. The universality germinates from our conventions, and it's our collective agreement on the meaning of concepts and words that allows us to assume their universality. But short of pure objectivity and true omniscience, it's not logically necessary for us to conclude or even presume that something known a priori contains universal truth.

"The universality germinates from our conventions"

The conventions have been invented, for sure, they are not in themselves a one to one correspondence with the nature of things, they are historical, but they do provide a scheme within which we do reach constructive and effectual positions and with which we have built practically successful sciences. This can't be doubted.


I would ask you, without a universal implication within the process of human knowledge, how then do you say we get things done? How is it possible to classify anything at all if what knowledge rests upon is a constantly moving picture without any real penetration by reason? Are you saying that knowledge itself is illusory? It seems incumbent upon you to pose some kind of answer to the question as to how knowledge can be possible, since you have merely negated thus far. But what you negate is a foundation for what we do take to be existence i.e. the advent of practically useful sciences.

--Pyth

[Aedes]

Originally Posted by Pythagorean

What I would propose is that the entirety of nature, whatever it is, is intertwined and therefore universals cannot be said to enjoy an independent status but must be unified, interconnected and somehow available to us through reason or as the reasoning process in action.

That is very well said, and that's how philosophy and the empirical sciences can unify, or at least complement each other.

I propose such a scheme because I think that if nature is said to be infinitely open regarding the production of natural 'classes' or types, then reason cannot, will not be grounded, cannot be said to posess real 'traction' at all and the ultimate justification for our knowing capacity is therefore illusory. Therefore, intelligibility or knowability must be limited not at the level of empirical nature (which is infinite and I believe merely probabalistic) but at the level of which our reason operates.

That makes sense -- though a psychologist would probably refer to cognition rather than reason (because I think cognition is a more neutral word -- reason is a bit more loaded).

I would ask you, without a universal implication within the process of human knowledge, how then do you say we get things done? How is it possible to classify anything at all if what knowledge rests upon is a constantly moving picture without any real penetration by reason? Are you saying that knowledge itself is illusory? It seems incumbent upon you to pose some kind of answer to the question as to how knowledge can be possible, since you have merely negated thus far. But what you negate is a foundation for what we do take to be existence i.e. the advent of practically useful sciences.

It's naturally easier to raise problems than offer solutions

But speaking as someone whose life is filled with science, I'm of the opinion that there can be degrees of knowledge and degrees of certainty that do not require unambiguous knowledge. In other words, life can be (and is!!!) practically lived based on assumptions of knowledge. That's why Descartes' thought experiment about the deceptive evil demon is simultaneously hugely insightful and hugely impractical. I might be deceived that I even have a body -- but come on, I can assume so without absolute knowledge that this isn't some dream. Besides, we all have a sense of propioception that knows our body is there -- so it's indeed empirical.

When it comes to knowing things, things beyond the repeatedly obvious (like that I wake up as the same person every morning), our requirements to be convinced vary depending on the importance of how that knowledge is applied. Obviously for medical practice the stakes are high so we need to verify knowledge empirically using our derivative of the scientific method -- and for the best studies we can document relationships (like between intervention and effect) stringently enough that no alternative explanation for that relationship can be logically posited.

In other words, knowledge is a functional thing. We dare to be wrong by living our lives without questioning everything. And for any given situation, the truly germane issue in the end is sufficiency of knowledge.

The foundations of all knowledge, or the supposition of universals, are interesting food for speculation. I'm not sure, however, that they're all that important in any practical sense. Science does what it does -- it hopes that its discoveries and inferences approach truth, but absolute foundational truth is not built into science. And many religions distrust the physical world in some degree or another. That's certainly true in Buddhism and in later Hinduism. But it's also true in Christian dualism in which the body and the physical can deceive, whereas truth lies in the spiritual. And this means that even if absolute truth is conceived of as revealed by prayer or meditation or faith, it means that all sensory observation is suspect and its truth remains hidden (i.e. the trees become lost in the forest).

We use the language of absolute truth, but it's used colloquially in the sense that for me as a doctor to say 'cigarette smoking causes lung cancer' does not mean that I ignore the limitations of empirical knowledge -- it just means that the knowledge is good enough that in colloquial use I can feel comfortable saying it firmly without disclaimers.

Far more interesting to me are the following questions:

1) How do we decide what to learn more about?
2) What makes us doubt?

[Pythagorean]

Originally Posted by Aedes

The foundations of all knowledge, or the supposition of universals, are interesting food for speculation. I'm not sure, however, that they're all that important in any practical sense. Science does what it does -- it hopes that its discoveries and inferences approach truth, but absolute foundational truth is not built into science.

It's interesting to note in this context that Plato's efforts toward an ultimate justification for knowledge centered around his search for providing stable political foundations for mankind, foundations which would allow thinkers to think freely and to allow human beings to live in a state of peace. As he wrote and which I will paraphrase here: 'Until philosophers become kings or until kings philosophize, their will be no cease of war and strife in the cities [nations].

It is also interesting to note how the sciences become tools for those seeking practical political or economic power and how they now behave in the consequential rearrangement of regimes from within. As a declining civilization loses its eros and drive to look for the foundations of knowledge and 'being,' the sciences then seem to lose their purpose and fall prey to those who no longer seek or share the same such ultimate justifications with all of its attendant spiritual baggage and social implications. And since a general inquiry is now subordinate to the practical uses thereof, the entire edifice loses its bearings, crashes and begins to burn up.

The practical questions such as your question: 'how do we decide what to learn more about?' cannot, in my opinion, be answered by way of a purely practical method. Since it was the philosophers of nature who originated the empirical method in the name of "truth", so I would argue, without their continued presence the empirical method naturally fails to furnish the rationale on its own. And so the trees only blur the wider conception and scheme of the forest; the quality of 'thisness' can not serve as a basis for the more important question of 'what is this?,' -an individual case can't be known unless we develop a wider scheme under which we might begin to categorize things and place it in a wider context and so life slowly becomes 'a tale full of sound and fury signifying nothing'. But again, the development of such a wider scheme would seem to require, as you point out, political and moral-religious obligations as a practical support. When we lose our nerve or faith in regards to making foundational generalizations we are left to squabble with one another in the dark, as it were, over which path forward we should take because there is no general plan. You might call this state of affairs 'post-modernity'.

Your second question 'What makes us doubt?' then, is not simply the absence of metaphysically grounded leadership but rather the belief that such leadership is innappropriate in our practical lives. It's not merely that we no longer know how the universal view complements the individual (which I believe it does), it's that we have developed a prejudice against it. And it is the prejudice par excellance because it removes the possibility that an ultimate truth could even exist. So there can be no higher appeal: decadence cum doubt is now institutionalized as it takes the place of a desire for a genuine philosopical inquiry.

...and philosophy is then dead. Which is already an old story. What it is is ultimately the religious and political story of a certain civilization which began with spiritual and metaphysical and practical and other inquiries and searches including the search for human liberty, and ends with money, greed and pornography, 'soft-despotism' and doubt et. al. Plato could have, a priori, predicted that such a thing would happen.

[nameless]

Originally Posted by Pythagorean

Ever Since Plato Philosophers Have been impressed by the fact that there are universal propositions which we can apparently know to be true with absolute certainty...

If the rest of your post is built on this sort of nonsense... It is an untrue statement from whatever angle viewed.
When you say that "philosophers have been impressed.." are you saying 'all'? Many? Some? Two? Sounds like a failed attempt at an 'appeal to popularity'; if everyone thinks or believes something, that don't make it so.
Science knows nothing to be true with absolute certainty. Nothing! That is the realm of fundamnentalist religion and zealotry, the 'delusions' of 'belief'.
Any science study will show you this, and show you why.

Also, calling something a 'fact' (which it is FAR from), again, don't necessarily make it so (another fallacy).

[hamletswords]

It is important to understand in exactly what sense a priori knowledge is "independent of experience." No philosopher has ever denied that a child has to learn that two and two make four by learning to count, and that the latter process involves contact with concrete objects. But this only means that without sense-experience one cannot acquire the concepts of number, in this case the concepts "two" and "four," that is, a child who has never learned to count, to associate different numerals with distinguishable objects, will not even understand what "two and two make four" means. What the philosophers who believe in a priori knowledge assert is only that once the concepts have been acquired, the proposition can be "seen" to be true by just thinking about it (by "the mere operation of thought," in Hume's phrase).

That is interesting to understand, because it seems to negate the entire idea. Priori is independent of experience, providing one has already had the experience, and had it enough times to properly ingrain it into one's thinking.

To me, it seems very much dependent on experience, and although conclusions may be pre-drawn on phenomenon one has no experience with, these conclusions are based on previous, related, experiences.

What is the deal with Plato thinking that priori knowledge was based on previous lives? In a sense, I can see it, in that most of everything we know (even back in Plato's time) was based on previous discovery, just not previous discovery and learning by a previous self (unless you really want to get all religious about it- either "all are one" or "every guy that made advancements in math was a previous incarnation of me, Mr. Big, The Man, Plato.")

I don't think that's what he meant, though. He seemed to think that everyone was born knowing 2+2=4 without any experience. I guess it was an exciting concept and he just got carried away with it.

Priori knowledge explains why my mom gets really nervous when she sees a black guy walking near her car. Only, that's true priori "knowledge", because she's never had any personal experience with a black guy breaking into her car.

Plato's quasi-religious exuberance with math bothers me. Sure, if you look around, you see alot of cool stuff created from mathematical calculations. But what's less easy to see is public policy based on more abstract math like polls, statistics and projections, and considering how often things go wrong in public policy, it makes me wonder exactly how useful is our love of math as a species.

Further, public policy isn't like the weather. Yes, it is complex, but with effort, it could be accurately and positively implemented. There's no need to use much effort or even ingenuity if statistics and projections are reliable. They are considered reliable. They have been proven to be unreliable again and again. But they are "easier" (which means nothing if they make things worse).

[Pythagorean]

Originally Posted by nameless

If the rest of your post is built on this sort of nonsense... It is an untrue statement from whatever angle viewed.
When you say that "philosophers have been impressed.." are you saying 'all'? Many? Some? Two? Sounds like a failed attempt at an 'appeal to popularity'; if everyone thinks or believes something, that don't make it so.
Science knows nothing to be true with absolute certainty. Nothing! That is the realm of fundamnentalist religion and zealotry, the 'delusions' of 'belief'.
Any science study will show you this, and show you why.

Also, calling something a 'fact' (which it is FAR from), again, don't necessarily make it so (another fallacy).

Nameless, I appreciate your strong feelings here. And I do admit that my very last post was rather tendentious (as well as being sloppy and poorly thought out ----sorry 'bout that, but it happens!! ).

But as far as the text that you have excerpted it was meant to eludicate the general problems related to a priori reasoning in philosophical propositions and arguments. They were meant to be purely pedogogical in nature. And by posting it I was just trying to stir constructive discussion.

As regards to whether or not human knowledge can be grounded in a form or forms of certainty I believe we should leave this question to philosophical type debates and not a priori preclude affirmative or negative proposals on the matter. Furthermore if science cannot prove anything to be absolutely certain then how can it state with absolute certainty that nothing can be proved? It seems to me a kind of paradox is at work here.

Be well, Nameless

--Pythagorean