Thales1940

In a recent post I defined cardinality and ordinality as having the property of being either exact or in-exact and induction starts from the context of the inexact. Consistent with Objectivist epistemology an analysis of the terms demonstrates 'exactness' as the conceptual common denominator. It is important to know that I am using ordinal and cardinal in a way that is more general than when defined by mathematics. Exactness has the same effect on the general problem of concept formation as does inexactness on ordinal numbers. Nature as perceived is always inexact. Exactness is a direct result of our ability to abstract and build exact concepts out of inexact perceptions. If you take a close look at Thales there is evidence to support my claim that he integrated things that were inexact into things possessing exactness. The process is identical to that of our integration of sensations into percepts. They both utilize a calculus, the difference being in the first case the mind integrates without the need for the exercise of choice, the second way introduces choice. We experience the sensational level of consciousness as a means of moving from a view of reality as an undifferentiated blur to that of distinct objects. Because of the fact that we perceive units it takes a focused analysis to see how units are derived from flux. There is no such thing as a pure abstraction. That is a mystics attempt to co-opt the exactness of a reasoned concept as a means of proving other-worldly-ness.

Objectivism teaches that to know an abstraction with certainty you must have taken the step of induced ordinal number integration. Because most everyone has ten fingers it is a common source for systems of numbers. There is a sense where we are said to have exactly ten fingers and there is a sense where that is not true. The relation between ordinal and cardinal numbers is where to find these two senses.

Objectivism assumes the integrity of perception. Kelley's book goes further. The point is that the greatest (an ordinal term) or widest abstraction is the axiom of existence as applied. That perceptions are always known in a context that is relational means that there is always more than one way to prove them. It is possible to relate two trees of different mass by observing that one is taller, the other shorter. There are a large number of choices open to mankind that are so obviously different as to not call for an argument. Further, this kind of knowledge which is pre-conceptual goes back to the beginning of the use of concepts.

Building from my last post, I want to 'chew' the ideas relating reason to emotion. Starting with the fact that nature has not automatically given us reason in the same way that it makes us hungry and horny, it follows that there is some key difference. Volition would seem to be the obvious choice. There is a sense where we choose to reason while emotions are just felt. One is active, the other passive. An important distinction among many, is that you have to figure out why reason is important. We don,t have to know why sex is important, that is taken care of by our feelings. Reason is a process that can analyze our emotions, our feelings and discover what causes them. There are reasons as to why we get hungry and horny. Identifying them gives you a power beyond the intensity felt before. This is a point worth inspecting. Music is a good way to illustrate the before and after of learning the reasons behind a given emotion.

If you are a person who knows what the 18th variation is to Rachmaninoff's Paganini Rhapsody and like it for no other reason than that it pleases your ear this is for you. About 1991 a Miss Martyn wrote a fine biography about Rachmaninoff. She explains that Rachmaninoff was interested in showcasing an integration between two forms of classical music, the piano concerto and sets of variations. The 18th variation is, without question, the center of the piece. It is the only one that turns up in music boxes. Rachmaninoff has an argument in support of the 18th, but most people love it on nothing more than their emotional ear. There are 24 variations in the piece, 23 related in obvious ways to the main theme. The 18th is different in a way unique to that variation. Twenty three variations are deduced, the 18th is induced. Rachmaninoff is teaching us a lesson in logic. The 23 are commensurable to the ear, the 18th has to rely on other tactics, it is incommensurable. The most obvious difference is that it is played twice, once by solo piano, and again with the piano supporting the orchestra. Less obvious is the fact that the 18th salutes reason by demonstrating an indirect proof, the base of all inductive knowledge. The 18th is taken from a reading of the original piece, upside down.

Sets of variations usually establish a theme and then go through a series of variations where the variation keeps the theme and varies the tempo, or some other aspect. In the case of the 18th, Rachmaninoff, by inverting the score retains the spacing (tempo) between the notes and creates a new tune. As with the other variations there is a mix of new and old with the mind capable of making the necessary but less obvious integration. It sounds familiar, and yet inexplicable. Knowing the why of the 18th increases the emotional response for those who love reason.

The same method of relating the causes of emotions in general, results in a greater reward.

This is a quote from Philip E. B. Jourdain's "The Nature of Mathematics' as reprinted in Newman's "The World of Mathematics', "Thales introduced the ideal of establishing by exact reasoning the relations between the different parts of a figure, so that some of them could be found by means of others in a manner strictly rigorous." Also, "The deductive character which he (Thales) gave to the science is his chief claim to distinction." Since Kant, the idea that the axioms of mathematics are the product of the minds of mathematicians has passed the idea that a cardinal number reflects gods perfection ala Pythagoras of 550 B.C. From Pythagoras to Kant, mathematics tried to find some way to relate the perfection of cardinal numbers to god and could not do it. Kant gave up the quest and ceded certainty to 'pure knowledge' in the form of 'apriori' concepts obtained without proof from nature. Thales integration as described by Jourdain above can not be justified by either method. Thales key to greatness is that he demonstrated a way to prove ideas without the need of god. He is completely natural.

The example History cites is in the fixing of the equinoxes with exactness. Before Thales visit to Egypt, circa 600 B.C., Egypt treated the certainty of mathematics as only true in theory. They noted that there is no way to apply a number to a thing exactly. It is this refusal to integrate the idea of cardinality to nature that leads to a slippery slope ending in skepticism. Egyptian skepticism is so complete that they could not figure out a way to measure the length of a years rotation with the sun. Using the Egyptian records which showed that there was a strong correlation between floods and the rising of the dog star, Thales discovered trigonometry. The objectivity of his discovery is underscored by the fact that the world now observes a standard calendar proved by trig.

Mathematics and logic are man's monuments to objective reason. Right now they may look a little abused. Most intellectuals today are Kantians in that they agree with Kant that the human mind is not acquainted with absolutes through perception. Absolutes according to Kant are dependent on 'pure reason' which means independent of perceptual reality. This makes it impossible to see the relation between any perceived 'first' and a conceptual 'one'. 'Firsts' belong to the world of perception and 'ones' are conceptual, abstract. Kant's inability to see how 'one' and 'first' are related goes all the way back to Pythagoras and the discovery that not all numbers are commensurable. All cardinal numbers are commensurable, because they are all exact and you can always combine exact with exact. No ordinal number is commensurable because no ordinal number has an exact value. It takes a calculus to make them exact. The calculus is how we prove induction.

A contradiction cannot exist, basic Aristotelian logic. Kant used a list of antinomies (contradictions) as support for his argument that the mind was limited to abstract certainties, unrelated to reality. Mathematics through the voice of Newton had just solved this problem. Newton showed that it is possible to integrate irrational numbers like pi to rational numbers like 1.2.3...where at the start they seem to be incommensurable.

Mathematics removed the contradiction with the calculus, the same is possible for epistemology, sort of. That is, mathematics does use a calculus to integrate rational with irrational numbers, but they don't understand the philosophical implications of their accomplishment. There is a famous quote from Einstein which embraces Kant's thinking, "Here arises a puzzle that has disturbed scientists of all periods. How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality? Can human reason without experience discover by pure thinking properties of real things?..."

"As far as the propositions of mathematics refer to reality they are not certain; and as far as they are certain they do not refer to reality." (Pg 340 of Kline's Mathematics, the loss of certainty). Einstein reveals his Kantian roots which make him a subjectivist when it comes to defining a cardinal number. A subjectivist in mathematics has no objective way to relate 'first' to 'one'.

Conceptually, the ordinal concept 'first' is among the most primitive. Einstein seems to be claiming that he sees no relation between 'first' and the cardinal 'one'. This is absurd on the face of it. Their relation is obvious to anyone who knows the meaning of both. For any given instance of 'first-nees' inspection will show there to be exactly one of them. The distinction between 'first' and 'second' is perceptual and is subject to all the problems of comparing perspectives. But, over thousands of years of experiencing ordinal numbers people began to generalize their attributes. The most important attribute of 'first' is that it always refers to just one thing. The concept 'one' refers to a generalization acquired from the continuous experience of 'firsts'. This means that Pythagoras was completely wrong in assuming that a cardinal number had some sort of mystical certainty, ultimately depending on god, instead of the generalized, induced knowledge from nature.

There is no history before Thales of people trying to determine the existence of god. It was taken for granted that he must exist. As soon as Thales loosed the power of reason on the world the most obvious question after Pythagoras is how do you relate reason and god, or cardinality and god?

More to come.

The discrete and the continuous-a paradox or antimony?
This project of certifying reason has kept my nose in my philosophy dictionary. I just looked up paradox and found the information interesting. I had always thought of paradox as a clear sign of an error in logic, which is impossible to generate. My dictionaries first definition is softer, something like, appears to be either true or false, but may be the opposite. It also says to compare this to antinomy. Antinomy takes a stronger position which denies integration, they name it extreme paradox. Together, these terms define the problem of incommensurables. A paradox only seems contradictory while an antimony is contradictory, they are incommensurable. Shucking to the core we find a comparison between our two logics: the logic of certainties and the logic of probabilities (deduction and induction) and, of course, they too, are incommensurable.

At this point it might help to recall that incommensurability was discovered as a problem by Pythagoras when pi's inexact nature would not seem to integrate to the exact world of the natural numbers. This problem was solved with the calculus and
so constitutes a paradox, but only after an extremely long time and the Pythagoreans always treated it as extreme, as an antinomy. It occurs to me that there are no antinomies, only paradoxes because we can always use a calculus to unify. When you apply a calculus to the seeming paradox of deduction and induction you factor in a small plus or minus to the inductive side of the equation making it commensurable. Referring back to the opening thought of Rand's IOE, she notes that concept formation is a process consisting of two similar methods, one based on differences and one based on similarities. When things are similar they are commensurable, when not, not.

Because of Rand we know that similar things can become the same through the process of measurement omission. Of equal importance is the method we use when we unify based on differences, ie, when they are incommensurable. Math teaches the lesson: you obtain commensurability by making some small plus or minus adjustment. Once commensurable you know what to do, unify by similars.

abstractions as the cause of cardinality
Thales is the father of philosophy, science and mathematics circa 600 B.C. Because - he did something that changed the way we look at the world. We know that Egyptian mathematics had a pure, theoretical aspect about it that made their cardinal numbers impractical. We have their 'rope stretchers' as an example of them using ordinal numbers to measure. I stipulate that the essential difference between an ordinal number and a cardinal is the degree of exactness. A length of rope blessed by a priest is more prone to change than a yard stick. Greek thoughts on numbers didn't go further than to see that they don't change, (the proof of cardinality). Thales considerable integration was to see that he could relate his cardinal numbers to the Egyptian cardinals and render them practical. This had the effect of showing Greeks how to prove their practical numbers and showing Egypt that their numbers can denote. Thales discovered it possible to overcome the problem of constant change. His demonstration was proved with reason and logic and is the first instance of a man proving a thought without referring to god. Whats the difference? The Greeks had, as a matter of course, assumed that numbers and examples of numbers were related while wherever mysticism was in charge the certainty of numbers couldn't be guaranteed. If god made them he could remake them differently. This is mysticism's way of denying objectivity. Science is not possible except by the relation of ordinal to cardinal number. Its as easy as relating 'first ' to 'one'. It is impossible to conceive of an instance of 'first' that does not imply an instance of 'one'. The way we prove an ordinal number is to point at it. In the case of rope stretchers, if asked the length of said rope, one has only to point and say "that long". The way we prove a cardinal number is to explain how it is related ordinally. Had there been a yard stick at 600 B.C. one could compare religious ropes to yard sticks and draw the appropriate conclusions of longer/shorter, etc.

In the history of philosophy an important question is the question of origins. If, objectivly, there was an origin to what we call 'western thinking', its objectivity makes it possible to find. Whatever the answer, it must be of a sort that anyone might notice. Philosophy starts with the obvious. Ordinal numbers are obvious. We usually think of ordinal numbers as being like first, second, third... which is not incorrect, just too limiting. Ordinal numbers are estimated measures made before we convert them into cardinal numbers that measure exactly. Part of my argument is that the difference between an ordinal and cardinal number is the difference of exactness. Since ordinal numbers are based on perception, in a sense they are subjective. Thales was the first to notice that the perception of 'firstness' could be the cause of the abstract 'one'. The time was around six hundred B,C, and everywhere that was relatively civilized both ordinal and cardinal numbers were in use. There are two descriptions of Thales that pertain to this question. 1., he is said to have estabished the equinoxes with exactness, a feat that requires knowledge of the integration of incommensurables and, 2., he is said to have discovered a way to prove the truth or falsity of geometrical figures without reference to anything other than logic and reason: god was superfluous. The first is related to induction, the second to deduction.

It was obvious to Thales that 'first' and 'one' are related causally, ie, every time you experience an event that qualifies as a first, by implication there is only one way to explain it. He saw that 'one' is a generalization abstracted from experience. In the case of 'one', the experiences are all instances of 'first'. Every change presents us with a new 'first' and more content for our generalizations. When the Pythagoreans discovered the value of pi they didn't know that abstracting the general direction of pi was enough to establish its identity. There is evidence to say that Thales understood ordinal/cardinal integration. An experienced number is an ordinal number like 'first, second, bigger, smaller, etc. These 'numbers ' exist' in the sense that the relations exist that give them content. There exists a relation between Indiana and Ohio that says that Ohio is larger than Indiana. "Larger' is an ordinal term that describes a measurable difference between the two states. Most people don't carry a large supply of measuring tools with them all the time. That means that the majority of the common measurments we make, each time our mind makes a comparison, are done using ordinal numbers. An ordinal number allows one to quantify differences without using cardinal numbers.

When a baby is born it has no ability to form abstractions. In the history of humans there was a time when humans did not think abstractly. So, obviously, there must have been a first time when someone made that first integration. Rand's theory of concepts describes how humans use abstract thought as a tool to enhance life. The first thing Rand proposed in her theory was to move the question of universals (which is what abstractions are) from metaphysics to epistemology. This has the effect of escaping the error of cosmology and correcting Pythagoras, he thought 'ones' existed in some sense independent of consciousness. Since this is exactly the opposite of the truth it caused the mind/body dichotoy in both philosophy and mathematics. It is easy to see in mathematics. Any good math history will tell you that a numbers certainty was based on axioms said to come from god. This didn't get changed until Kant when a numbers axioms were said to apriori, based on a concept pure of existence. When Pythagoras is corrected the figure of Thales should have caught her attention. There are no 'ones' in reality, but there are 'firsts'. To form the abstract 'one' is a piece of cake if you know what 'first' means and Thales knew.

"First' is the most common word that comes to mind when someone asks you to explain ordinality. Even if you don't have a concept to describe 'first' your life will still experience firsts constantly. A cave man experienced a life full of 'firsts' without the help of 'ones'. What is the difference between a percept and a concept? It is the same as the differene between 'first' and 'one'. Thales didn't discover 'one', he discovered how 'one' is related to 'first'. By relating them Thales gave a way to transform some particulars into generalities. If you generalize a series of 'firsts', each of which is different, you can create an abstract name that denotes 'one' by noting that each instance of 'first' implies a 'one'. The fact that a caveman is the first of his mother's children does not require that she know what 'one' is. Yet, until my brother was born my mother had only one child. All firsts are different, all ones are the same. Remember, Rand begins IOE with the fact that abstractions involve a process consisting of two methods, each of which is capable of proving an abstraction. One way compares things that are different, the other, things that are the same. The bulk of the rest of Rand's theory focuses on explaining how similarities are integrated to form concepts. In fact she proscribes the ability to form a concept starting with incommensurables. If it is true, and it is, that similarities can only be integrated to similarities, then how is it that differences work?

Ordinal numbers are more basic than are cardinals because cardinals are integrated ordinals. Our life is a continuum of firsts rendered discrete by noting an abstract nature. There are no cardinal numbers in nature except as the product of an integrating mind. Obviously, if nature contains no cardinals in the raw so to speak it must contain something that can become a cardinal number. These would be your ordinals which are of course, by definition, incommensurable. My oldest brother was my mothers first child and first boy. My sister was my mothers first girl and fifth child. There are no two things about my brother and sister that are identical and yet they share 'first' as an attribute in exactly the same way.

Proofs, direct and indirect
Continuing my attempts to bring some kind of certainty to philosophy, this is an argument that unites perception and irrational numbers in that they are both ordinal and can only be proved indirectly. This depends on Rand's correct identification that the concept 'unit' is the base of all numbers and concepts. A 'unit' is conceptual. To see the difference between perceptual and conceptual just compare men to apes relative to the ability to integrate things into units. We (men and apes) see the same/similar world. 'Same" is a concept integrated from perceptual things that are similar. Thousands of years ago men knew what it meant to have two sons, two feet, two eyes, leading to the conclusion that "two" has an identity based on inductive evidence. We hold the abstraction absolutely because until put to use it can harm no one. The 'putting to use' is an example of mind/body integration. The abstract 'two' is proved every time units are united and has no theoretical meaning separate from experience. The practical result of Pythagoras is that mankind lost the ability to prove their numbers

Just over five years ago a man challenged me to prove that Rand is right. Implicit in the request was the idea that I could do it. You (kind reader) need to know that I was a 62 yr old gay male and the man was 25 and straight. My first chore was to relate reason to force so that he cold be sure this was safe to do. The problem with abstractions is that people can be dishonest. You might say that we negotiated a pact which said that we would discuss abstract ideas and that ideas, until used, are harmless abstractions. Take the abstraction 'two'. "two' has no particular existence, it derives its meaning from my knowing that I have two feet, two hands, two eyes, two parents, drawing the conclusion that 'two' means each of those integrations. Now, I can kick with my feet, hit with my hands, etc, but I can't DO anything with the concept two. 'Two' only becomes problematic when applied to pairs that are real. Rand ends the opening essay in "For the New Intellectual' with the naming of two conditions men must meet to gain trust. You have to promise to argue with cardinal (proved) ideas and you have to promise never to resort to force instead of an argument. Your integrity is measured by the facts that follow.

To the extent that I have successfully implemented the values of John Galt I have become visible to those who know those ideas whether they know who John Galt is or not. If I introspect and recall the relations in my life, the most significant have been those where reason was the implicit basis of the relation. What I have done this time is to make the role of reason explicit. Explicitness evokes a stronger emotional response, there is a relation between the power of your emotions and the reasons that support them.

Two unfinished ideas that I will try to finish later. think Kate will like them already.

Stated on my home page is a goal to teach the world the value of reason. The difference between objective and subjective in epistemology is that one can be verified and the other cannot. Integrity is a concept that refers to the integrations required for conceptualization. In general the concept furniture involves relations between tables, chairs and other household furnishings. In particular, as long as you don't contradict household furnishing relations you are guaranteed epistemological integrity. In every case it is always possible to compare the concept to the things in its class and thus verify it. It is as precise as double entry bookkeeping.

The certainty of objectivity is derived from a real relation that can withstand criticism. If I claim to have furniture and I, in fact, have some tables and chairs for proof, my position is superior to the subjectivist. Take, for example, any mystical claim of otherworldly relations. The first thing you have to see is that the claimed relations have no objectivity. The subjectivist starts with the premise that to be is to not be relative. This changes consciousness into a metaphysical causer instead of identifier. If god is 'the absolute" the concept is purchased at the cost of verification. It is not possible to verify a bookkeeping question if you only have one number to work with.

Humans existed for an inordinately long time before they discovered there was such a thing as mind/body integration, ie, that there exist a way to relate abstractions to concretes. It is a category difference similar to the difference between ethics and meta-ethics, one studies the thing while the other studies the idea about the thing. Before Thales, no one had pointed out that the universe is knowable, while, all the while, one hundred percent of the inductive evidence supported the idea that the universe is knowable. Abstractions and concretes are catagorically different, they are incommensurable. Anytime that you can honestly relate empirical evidence with the concept 'all' or its equivalent you can claim certainty. Using the same inductive data Pythagoras, on Thales heel, reversed Thales progress. "The oriental mysticism of Pythagoras, however, reversed this state of affairs and gave to mathematics a supra-sensuous reality of which the world of appearances was a counterpart." Boyer's hist of the calculus. Thales and Pythagoras were both focused on the idea that numbers could be used to prove things. Thales unified his deductions (proofs) on empirical (natural) evidence while Pythagoras did the opposite.

Numbers are so basic to our thought process that the analogy of a fish and water comes to mind. Rand has explained how mathematics and epistemology are related in her theory of concepts. All concepts are based on the idea that the mind can abstract from particular things which have particular identities by relating 'thing' to 'unit'. All concepts consist of real units whether we think of them as things or units. Things exist in relation to other things and all relations are messy, ie they have more than one perspective. A unit is a blend, a unification of two perspectives which only implies their respective differences while naming what is the same. There is nothing subjective or other worldly about this process, it is all objective.

The proof that exposes Pythagoras as a fraud is his inability to understand incommensurablity. Incommensurability is basic to concept formation because the first integrations are inductive in nature, ie messy. The fact that there exist both rational and irrational numbers and that we have learned how to integrate them can be used as proof for either mathematics or epistemology, it depends on your focus. Cardinality applies to concepts in general, exactly as it does to numbers in general.

Mathematics and concept theory
The simple difference between induction and deduction is that one argues from the parts to the whole while the other argues from the whole to its parts. It is the same process viewed from two perspectives, they both perform the same integration with only the direction changed. Consistent with Rand's view that existence is prior to consciousness, induction precedes deduction. This brings up an interesting question: How is it that Rand solved for deduction and not induction? Induction has to be understood before we can tackle the problem of deduction. I think that Rand did solve induction in practice but not in theory, whereas, for deduction she did both.

Rand uses many concepts in support of her theory of concepts that have equal application to mathematics, the most obvious being 'unit'. All of mathematics is an extrapolation of the concept 'unit'. All of concept formation is an extrapolation of the concept 'unit'. Any given unit has both a quality and a quantity. When Rand puts the concept of 'commensureability' to use in her epistemology she was using the term in a way that does not contradict its mathematical meaning. In mathematics two numbers are commensurable when they share a common unit of measure. In conceptual theory Rand identified two concepts as commensurable if they share a conceptual common denominator. In mathematics not all numbers are commensurable and the same applies to concepts. There is a difference at this point in the history of ideas where mathematics and concept theory separated and it has to do with the idea of incommensurability. Consistent with the idea that induction must provide the content of deductive reasoning we can infer that Thales had to have at least a practical sense of induction's efficacy because he introduced the world to deductive reasoning. The relation between percepts and concepts supports this position.

The problem of incommensurability was discovered as a problem by Pythagoras or a Pythagorean at the dawn of western thought. Thales, the father of western thought has dates that end where Pythagoras begins, some claim there was contact. However, the Pythagoreans get credit for bringing rigor to mathematics. In the process it was discovered that a certain kind of triangle could not be described with just the natural numbers. They meant by that that there existed numbers that described exactly and those that did not. This is the earliest attempt to explain cardinal and ordinal numbers. They called the latter irrational, (unrelatable). Rand identifies the same thing in concept formation as the attempt to mix apples and oranges. Mathematics developed the calculus and analysis and finally has a way to explain how to integrate incommensurables. We do the same thing with apples and oranges when we class them together under the wider concept 'fruit'. This means that deduction depends on commensurability while induction depends on incommensurability. Later I will show where Rand integrated incommensurable concepts which is her practical contribution while explicitly claiming the opposite, her theoretical gap.

Ordinal and cardinal as applied to numbers describes the relation between 'one' and 'first'. Since Rand identifies the concept 'unit' as the bridge between metaphysics and epistemology any concept that applies directly to 'unit' deserves attention. 'First' is relational and depends on a context for us to know the meaning. Ordinal numbers are always applied to particulars. Cardinal numbers are generalized ordinal numbers. Cardinal numbers are induced from our perceptual phenomena in the same way that fruit is induced from apples and oranges. There are no cardinal numbers in nature, there are only quantified relations.

When I first posted an argument on the net, I had in my mind that while the world's idea of philosophy is explicitly against me, implicitly the world is on my side, its just a small matter of showing the entire world that they are smart when they think I am saying they are stupid.

Take my example of the calendar as a probable choice as to where objectivity was first proved. Half of the world is explicitly opposed to 'objective answers' if they contradict their idea of God. Now these are mostly people who lead reasonable lives, they expect their bank account to balance. It is easy to prove that any 'other-worldly definition of 'objectivity' fails from its non-objective starting point. The objectivity (practicality) of such concepts as 'number' and 'the law' are natural and this-worldly.

The other half of philosophy is explicitly opposed to the idea of 'objectivity' which means explicitly they hold it impossible to prove in court a right and wrong distinction involving objective numbers, ie, numbers that don't change for either party. Virtually no one lives their life acting on either of these two subjective errors. Notice that the error involves a one-sided perspective on change. The mystic is dogmatic, holding absolutes (things that don't change) as other worldly. His opposite is the opposite of dogmatic, claiming absolutes are impossible, you can't step in the same river twice.

Objectivity proves that there are some contexts that can be proved and some that cannot. Proof depends on the context, proof depends on the integration of the real facts (body) to the concepts (mind) that describe them.

Since the facts are on my side what we have is an argument over the meaning of words, abstractions, concepts. I have seen reference to some sort of dooms day clock which purports to be measuring how close in time we are to our own destruction. It is very close to midnight if either of the subjective ideas are acted on and back to zero if Thales' value is discovered. The absurdity is that two subjective positions are giving objective reasons (?) as to why we should believe them. Neither side is willing to argue from the honest position which is that they don't use reason in this kind of argument. They give reasons for doubting 'reason'. I am the one who professes to love reason.

Polemics is driving the world to extreme positions. The only escape is to rediscover the answer to what is called the Greek miracal based on the facts surrounding Thales and six hundred BC. Take another look at my first post.

This is an argument to establish the relation between what is commonly called the real and the ideal.

Sciaberra in his radical Rand book notes that: "Lossky characterized his intuitivist philosophy as an integration of idealism and realism.", pg 45. Lossky was Rand's philosophy prof in university. Rand's unification of the mind/body dichotomy which she related back to Pythagoras accomplishes Lossky's dream. Mind/body integration comes from the unity of the real with the ideal. As with all fundamental integrations, the unity of ideal and real involves the problem of incommensurables. There is no common unit of measure between the real and the ideal, they are as different as ordinal and cardinal numbers, one perspective can be integrated to change, the other cannot. Things change, once unified they don't. My finger is changing as I gaze at it, blood is in motion, nails growing. When I think of my finger in the abstract, within that context there is no change. OK, I'm playing with you, mathematics has solved the problem of how to integrate ordinal and cardinal numbers and this is the same problem so, if we apply the same reasoning to the concepts 'real' and 'ideal', philosophy will finally reverse Pythagoras.

When Pythagoras reified numbers he effectively reified 'units', ie, confused the thing with its idea. Consistent with Rand's view that a concept and a number are both based on a unit when you attribute something to one you effect both. You may only know about the one, but the hard fact is that you get the effect on both. In the case of Pythagoras, mathematics realized they had a problem, epistemology did not. To be fair, the world didn't have explicit knowledge until Rand, but they had the same evidence. When Egypt taught that the calendar year was three hundred and sixty days long it was efficient for ease of calculation but wrong. Adding a five day festival at the end of the year was inductive progress, but over four years still wrong enough to miscount by one full day, more or less. Leap year is an idealized conceptual adjustment to render an ever changing relation into something that holds still long enough to use it.

Integration of the real to the ideal
This is directed to those Objectivists who have an integrated view of Rand's theory of concepts. Because of Rand we can know that a concept denotes exactly in a universe where consciousness perceives relationally, ie, inexactly. We generalize from the particulars we see. The explanation as to how these opposites relate explains certainty from either an inductive or deductive context, certainty is contextual and it works from both directions. The problem with the world is that philosophy is split between these two perspectives with each arguing for total isolation from the other instead of explaining how they are related. The fact that 'change' is both a concept and a fact is unknown in the world of philosophy because facts are seen as induced particulars and concepts are seen as perfect abstractions devoid of any factual content. In general all relations can be viewed from either of these two perspectives, ie, before the integration or after. Before certainty and after - through induction, through deduction - as a part, as a whole.

We can treat any new blend as one thing and note its attributes. For instance, we can take a horse and a jackass, mate them and produce a mule which has its own problems. There are attributes of a mule that neither horse nor jackass have. This is nature pointing to how a concept is made. I don't mean to be anthropomorphic, I mean nature is exhibiting how to integrate incommensurables. It is our type of consciousness that has to weigh incommensurables. To feel is to weigh. Stop and think for a moment about some of the many weighs your body is accomplishing while you are busy at the computer. My butt is on a swivel chair where my body has automatically adjusted to the variables observed through touch and hearing mostly, so that I don't have to worry about falling. It is our type of consciousness integrated to existence that is the ultimate incommensurable integration. Consciousness and existence have no common unit of measure, but they are similar. How does Rand's theory of concepts fit with the above? By omitting dissimilar measurements Rand treats two things as one. We can always reverse the process and reweigh the evidence. We know that you can't add apples and oranges but you can add fruit. She explicitly restricts certainty to commensurables. This does not integrate to Rands own words as told to us by Binswanger in the second ed. IOE, pg 190-196. Rand makes clear that her certainties depend for their content on induction when she uses a calculus to integrate the continuous to the discrete. Objectivism, much less philosophy in general, has not realized that Rand solved the problem of induction. The world of philosophy views Rand through the eyes of subjectivism and so cannot explain Pythagoras. Mathematics is more clear on this point, esp Boyer.

reason and implication
My argument goes something like this: Reason is necessary for survival. If you had to list the order the things necessary for survival, reason has to be number one. Has that always been the case? Well, no, there was a time before we developed abstract thinking. Reason itself is an abstract thought. There was also a time between when we first developed abstract thinking and when we first knew we had developed abstract thinking. For instance, people associated the index finger to the idea of 'one'ness long before they developed names for numbers. The same is the case for reason. People were reasonable long before they figured out what that meant. Its the whole idea of implication. I am implicitly telling you by pointing my finger at you that I mean you in some way. Imagine a scenario where a boy has his back to his mother and is intently focused on catching a fish. She wants his attention, makes a noise and points at him. You could even have the boy shrug to convay, 'Who, me?.'. Given similar contexts the same scenario can transcend centuries. Buried in this is the explanation of the connection between things similar and things the same.

The idea of unity is everywhere in nature simply because our perceptions use unity as an organizing tool. Our nature is to apprehend nature one unit at a time. We can't change our nature, the given is that we feel --- things ('we feel things' is a good example of mind/body integration). We can help or hinder the conceptual level. Percepts give us units, concepts measure them, tells you what they are, You cannot question that you feel, you can question what it means, or is to feel. Reason focuses a critical eye on perceptions. Perception gives you the facts, a reasoned epistemology identifies objectives. When I point my finger at you the objective is to unify the idea of you with the idea of my finger. I think of my finger as a unit and I think of you as a unit and it is possible to relate these two units. That's easy on the perceptual level, that is, we can see my finger and we can see you, but what about some more abstract idea? An easy second level abstraction would be 'furniture'. 'Furniture' is about particulars but is general. Add one table to one chair and you have two pieces of furniture. Of course we can still point to the table and the chair and mean the furniture.

What about 'integrity'? 'Integrity' has a long chain of abstractions that relate back to the idea that everything that is real is related to every other thing that is real., the rest is a matter of mathematics.

Objectivism
I have read somewhere that Rand preferred existentialism as a name for her philosophy. Given that she argues for a primacy of existence position I can see her point, yet look at all of her opposition, it is all subjective. My arguments are aimed at the professed subjectivist, the mystics won't admit they are subjectivists. A professed subjectivist is one who holds conceptual certainty as an impossibility and so denies the veracity of deductive reasoning. Thales is the first to demonstrate deductive reasoning, obviously he did not practice the subjective limit, he is the first objective thinker.

Consider Thales context. The Egyptian world he found had no critical thinking. The Greeks commonly assumed that numbers used in trade gave exact measure, not so Egypt. This difference is what got Thales to thinking. He discovered that one system allowed for verification, the other didn't and that is a significant difference. It is as if he discovered the value of double entry bookkeeping. Using Egyptian records of solar activity and sticks to make shadows Thales fixed the equinoxes exactly. Since the relation between sun and earth demonstrates an irrational number, the only way to achieve that kind of certainty is indirectly. Thales, by fixing the equinoxes demonstrates the relation between inductive experience and deductive certainty.

Deduction argues from the perspective of the whole, a concept impossible to a subjectivist. Both subjectivist and objectivist start with the same perceptual evidence, the difference is to be found in the conclusions drawn from the evidence. Assume there is an objective reality. Does the idea of an objective reality integrate with your experience? The subjectivist says no because experience gives only part of the picture. The objectivist embraces the subjective position as far as it goes to establish experience and then using said experience establishes objectivity by an indirect proof. It is true, there is no direct way to experience the absolute, so, there must be an indirect route, and Thales was the first to find it. Because it is based entirely on inductive evidence it is not possible to be consistent and dogmatic. If there is a question, one always goes back to the proof, which is the experience. There is only one way to relate experience to proof and that is by the indirect method. There is only one way to know the whole thing and that is by knowing all the parts. This is the identical problem faced by the Pythagoreans and their discovery of incommensurability between natural numbers and irrationals, or one could say between cardinal and ordinal numbers.

Rand implicitly solves this problem when she explains that it is possible to integrate the discrete and the continuous by a small plus or minus adjustment when needed. Thus she uses a calculus in epistemology in the same way that Newton does for mathematics, and this integrates exactly to Rands idea that mathematics and epistemology share the same base: the unit perspective.

The relation betwen the part and the whole
When we contemplate about the way we think (introspect), we can discover that there basically are two ways to do it. The first way describes how we develop thought from infancy, the second describes a way to prove the first. A baby has no grasp of 'the whole' and is a little empiricist fire ball. As we mature we gradually begin to draw conclusions about things we have experienced. This is natural 'mind/body' integration. Induction applies to the first way, deduction, the second. In exact relation to induction/deduction are our ordinal and cardinal numbers. Boyer in his history of the calculus notes properties of induction as being ordinal and relational. There was a time in history when Egypt thought ordinal numbers, in the sense given, the only way to measure real examples of problems. They did not know how to relate a theoretical 'pure' number to the practical, messy real world. They suffered from the mind body dichotomy before Thales introduced deductive reasoning. Deductive reasoning requires mind/body integrity, a breach renders exact conclusions impossible.

In today's world there are two dominant answers to the question, "What constitutes an exact conclusion?'. Both are subjective and so, unprovable. If you take either of these ideas as foundational, science and the golden age of Greece could never have happened. If the golden age of Greece was, it must have been caused by some other method. Best evidence says to look to Thales. Among the actions attributed to Thales the most important would seem to be philosophy's assessment that he was the first naturalist, he looked to this world for answers. This is in opposition to all previous mystical whim. Implicitly, since we know Thales is neither mystic or Kantian subjectivist, it would make sense to ask if he could be objective. Objectivity can be defined as the integration of a cardinal number to its ordinal content. Consistent with our two ways of thinking, one is inductive, the other deductive. One integrates incommensurables, the other commensurables, both can indicate certainty.

Have to take a break.

More on 'certainty'
I just read a post where the poster identified himself as a tenured philosopher who didn't know what philosophy was. This person is no absolutist. I did see a post by an absolutist, but his absolute requires you transcend this world to find it. Neither can define a 'this world' context for certainty. An absolute is a defined context. As someone once said, "All knowledge is contextual". What is a context? One answer is that it is like the concept 'limit' as used in calculus. Calculus is a term that identifies the world of change. If we didn't limit said world of change we would never be able to render it practical.

'Least' is an ordinal term that describes a trend towards zero as close as we please. 'Most' approaches 100%. The numbers can be replaced by the ordinal 'none' and 'all' without losing any meaning. A practical demonstration of creating a context might be: I have five fingers on my right hand and this can be verified by simple perception. At the same time my fingernails are growing. Or, I just cut my nails. Are we talking about the same fingers before and after the cutting of the nails? Context sets the limit. If the context is about nail cutting it matters, if not, not. If someone were to ask how many fingers I have and I say five with some small parts in the trash I would be mixing the contexts. The other view might be that I am a 'hand model' with perfect nails. In that context the nails make a difference. A context gives us a way to organize inductions. The moral about the seven blind men and the elephant is about the importance of context which in that case was the whole elephant. A part is not equal to the whole as any practical demonstration proves. Cantor's ideas about the actual infinite have no context and are not practical. The man who defined ordinal and cardinal didn't know what they mean. A cardinal number is established through the method of rendering an ordinal number general.

It is obvious to most Americans that we, as a nation, are becoming divided in a way that is trending towards disaster. This divide is fundamental in that it goes to the very core of two radically different ways of justifying life as a subjectivist. This translates immediately into a disagreement as to the meaning of war from two subjective positions. BTW, our kids "feel" this, but don't have the arguments to explain their black clothes.

My argument names these two the religious and the secular. In philosophy that generally means the rationalists and the empiricists. Implicitly they are the asserted cardinalists and the asserted anti-cardinalists, neither backing up their assertion - they are subjective.

A cardinalist is one who asserts a relation with the "absolute". An anti-cardinalist is a relativist who rejects the idea of "the absolute" all together. That, by implication, makes him/her an exclusive ordinalist.

Imagine a see saw that, in theory, represents everything asserted cardinal at one end and everything asserted anti-cardinal (ordinal) at the other. Or, because I have already established a relation between ordinal/cardinal and relative/absolute, we could have a see saw with relative/absolute as the extremes.

From Pythagoras to Kant, there is only one fundamental answer to the question: how is it that cardinal numbers describe perfection when perception doesn't know what that means?. Pythagoras said it was because numbers had a "special" (read "non-relational") status, which meant that the reason was beyond the mind to know. This makes a practical solution impossible. When Heraclitus says that everything is change, he can't integrate absolute change to cardinal numbers because THEY don't change. That cardinal numbers don't change then becomes the basis of Parmenides' argument. The mind/body dichotomy is the same as the relation between practice and theory. By explaining the relation between practice and theory it is possible to rid the world of the mind/body dichotomy. Cardinal numbers are the theoretical child of ordinal numbers; they are conceptual. Ordinals are derived from experience, they are perceptual. Ordinals are practical, cardinals are part of the theory that sometimes ordinals can lead to cardinals. As in the case of recognizing that the ordinal "all" is identical to the cardinal "100%".

Philosophy, from Pythagoras to Kant, is a history of the refusal to look at the mind/body problem except from one perspective, the subjective cardinal, (god is all). A subjective cardinalist is one who claims to know where cardinality comes from, but has no evidence to back the claim. Kant, in an attempt to save the subjective cardinal, compares it, for the first time, to the subjective ordinal.

Hume had pushed Kant into the role of trying to save causality without giving up god. No one had been able to figure out a way to integrate them. Kant came up with the first fundamental change in 2400 years. He said the integration is impossible, from our subjective perspective. By that, Kant was implying that god has both an ordinal and a cardinal perspective but we are only aware of the cardinal, and we don't know how. Perception is not perfect and so can not be integrated to god who is equal to perfection. Kant's "pure knowledge" is an attempt to make gods perfection available by an end run around perception.

The first thing that did was to make Kant walk up the see saw and reside alone at the anti-cardinal end, making him the first secularist. Gradually though, people started to look at his idea and see some improvement over the subjective cardinal. For one, it was not dogmatic, well, except for one thing, it had no starting point, no cardinality at all. That is the dogma of the relativist.

Still, with no Rand, and a misunderstanding of the meaning of Thales, seemingly "no dogma" was better than blatant dogma. Fifty years ago the church seemed unchallenged, but was in fact under severe attack in the philosophy and math departments all around the world. 2000's election gave us proof that Kant has caught up and the see saw is balanced. Four years later only strengthens that argument. The fact that there was no significant change supports the idea that as a nation, and maybe as a world, we are stuck between two opposites that because of the dogma of each, can see only their own perspective. The biggest difference between the two is that the subjective cardinalist is a liar, the anti-cardinalist is just mistaken.