Wednesday, February 01, 2006

Axioms and Irrational Numbers: Philosophy vs Science

The other day, in my History of Philosophy class, we talked about Irrational numbers. Such a peculiar name for numbers as such, isn’t it? I suppose it’s to reflect our disposition toward these numbers, in that they make no sense to us, hence irrational. Anyway, the discussion about Irrational numbers arose because the professor was trying to argue about the duality of things – at least some things – duality in their nature, as having properties of one and many at the same time, or of being and non-being at the same time. I suppose this discussion was apt in the context of Plato’s Theaetetus that we were reading that tackled the perennial question of “what is knowledge?” Is knowledge wisdom? Is wisdom that by which the wise become wise? So, Irrational numbers – according to their nature – exhibit a certain duality of sorts. They can never be expressed numerically as a distinct and discrete quantity. Yet, one can illustrate an Irrational number in figures and diagrams with discrete boundaries and measurements. Hence, the apparently confounding nature of these numbers that render themselves entirely distinct in one feature, but utterly boundless in another form is, to say the least, very puzzling, and very interesting. In class, I thought of how interesting a parallel these Irrational numbers have with newer discoveries about the nature of Quantum particles. Based upon latest scientific knowledge (which may or may not be changed in the future), it seems like quantum particles exhibit similar “irrational” dualities of have different natures of "being" in different situations. To the novitiate Objectivist, this new revelation of duality in existence comes as a shock and repudiation of some deeply held convictions. Does this mean that the Aristotelian principle of non-contradiction and the Objectivist axiom of Identity have been invalidated? On the surface, that is what it seems. However, these doubts and hasty conclusions typically have at their root, a fundamental misunderstanding – or maybe a complete lack of understanding – of the nature of axioms and the role of philosophy. Take water for example: The atomic elements that constitute what we call "water" remains the same regardless of what state of "being" we observe it in - water can be solid, liquid, or vaporous. The apparent existence of discrete boundaries when water is ice and the lack of it when water is liquid or vaporous has no bearing on the fact that water exists with an identity. One of the most illuminating insights of Objectivism is that causality is an expression of an existent's identity, i.e. an existent's identity determines its actions; the law of causality is not attempting to explain the action of one existent over another but of the existent's nature and its actions/manifestations. So, regardless of what new scientific discoveries tell us, we must understand that once we have a set of valid axiomatic principles, they cannot be disproven, changed, or modified by any new bits of knowledge. In fact, all knowledge should consistently conform to the fundamental axioms, and that is one good way of fact-checking. The role of philosophy is not to constantly change and adapt its principles with every new wave of scientific knowledge. This has been a great error committed by many philosophers of the past. They have seeked to have their philosophical theories corroborated by Science, rather than observe the dynamic and reciprocal relationship between the two fields. Philosophy certainly provides the most expansive ground of principles for Science to build upon, but above that, Philosophy relies on Science for inductive principles and Science relies on Philosophy for logical methodology, insights into nature and identity, etc. As Rand said, Philosophy says that things exist and that they have specific natures; it is now the job of Science to discover the specific identities of these existents, classify and categorize them, and build a hierarchy of information. In his book, the Russian Radical, Sciabarra explains Rand as seeking a "reconciliation of philosophy and science." Rand had argued that "Philosophy cannot depend on a changing physics for its ontological foundations... [but] genuine science must depend on philosophy to validate its modes of inquiry." Quoting further from the Russian Radical:
"...cosmological speculation depends on an imaginary omniscient standpoint. As Peikoff emphasizes, Rand's Objectivism makes a distinction between metaphysics and fantasy. There can be no purely deductive attempt to reveal the ultimate substances of reality."
When Einstein broke out with this theory of Relativity in Science, there was a flurry of activity in non-scientific circles to emulate Einstein's brilliant theory in their respective fields - thus, a culture of relativism gained influence in the Arts, in philosophy, in anthropology, etc. Before that, Newtonian physics and possibly also Darwinian theories of evolution influenced philosophers into rigid reductionism, atomism, and determinism. In Evidence of the Senses, David Kelley makes the argument that there are also Galilean influences in Cartesian, Kantian and Lockean theories. According to Kelley, the subjectivist and representationalist theories of consciousness borrowed their credibility from the scientific discoveries Galileo made in studies of perceptions of color and temperature. So, going back to my philosophy class, the professor discussed the "problem" of Irrational numbers in such a way that I think left an impression in the minds of the students that reality is in a state of flux. Things are neither this nor that but can be both and not both. Contradictions are a part of reality. It is easy to see then how mysticism and supernaturalism can easily creep in under such an unruly and chaotic epistemology.


Blogger Adrian Hester said...

Well, since I didn't hear what your professor actually said, I can't comment on that, but there's nothing particularly odd about irrational numbers. They're called irrational simply because they can't be written as a ratio of integers. The proof that there are irrational numbers (specifically the square root of two) was a great blow to the Pythagoreans, who saw the integers as the essence of reality; specifically, it meant that there was no finite geometrical procedure for measuring an arbitrarily chosen length. (You can get ever closer, but with only a finite number of geometrical constructions you can't get better than an approximation to the length of the diagonal of a square in terms of one of its sides, for example.) But that's really no more significant than the fact that you can't trisect an arbitrary angle with compass and straight-edge alone in a finite number of steps, which is a blow to a certain type of approach to pure mathematics but has no major implication for practical activities--you can measure it as close as you need for your purposes, and it doesn't matter whether the length you measure is a rational or an irrational multiple of your measuring unit. What it means for arithmetic in a decimal system is that not all numbers can be named or specified in a finite or closed form. They can still be handled just like any other number, they just can't be easily named in a decimal system.

2/14/2006 12:17:00 PM  
Blogger Ergo Sum said...

From your comments on my "Godel's Insight" post and this one, it seems to me that you subscribe to the idea that the mathematical realm of pure reason, pure ideas, pure logic, is separate and have little or no impact on actual, practical, or concrete realities and events. I suppose it could be characterized as a Platonistic dissection of the Ideal realm and the realm of real concretes (what Plato would have called "shadows").

I'm trying to understand why and how this is possible, or if it infact is possible -- this separation of pure mathematics from concretes. I wish knew more about math.

2/14/2006 04:41:00 PM  
Blogger Adrian Hester said...

No. Mathematics is the study of abstract relations, and mathematical statements are taken as true or valid when they can be shown to follow by deduction from other valid statements, and ultimately from axioms. However, it's an empirical question whether a given mathematical model applies usefully to a given part of reality, whether the axioms actually express something significant enough about that part of reality that it's worth following up the consequences of the model by experimentation and observation. The Pythagoreans took a rationalistic approach to mathematics, considering integers to be the source from which all else flows; the existence of rational numbers shows that this approach is inadequate--the integers are simply too sparse a set of tools for all numbers arising naturally in the process of measurement. This doesn't invalidate number theory concerned largely with numbers; it certainly doesn't mean that mathematics is in some way logically detached from reality; it means that integers aren't enough for all the things you might want to do with them, and it certainly means that a Pythagorean integer-mysticism, which heavily influenced Plato, doesn't even succeed on its own terms (legend has it that the Pythagorean adept who discovered the proof that the square root of two is irrational was put to death as a consequence).

Furthermore, you write, "it seems to me that you subscribe to the idea that the mathematical realm of pure reason, pure ideas, pure logic, is separate and have little or no impact on actual, practical, or concrete realities and events." Hardly. It's a simple matter of fact that the existence of irrational numbers doesn't pose insurmountable difficulties for such things as measurement. Many mathematical statements have great significance for practical actions, and many others do not, as most any scientist can attest with a wealth of examples; the point is that the existence of irrational numbers doesn't matter much for day-to-day life, though they have great mathematical significance.

2/14/2006 05:35:00 PM  
Blogger Ergo Sum said...

You said:

"The Pythagoreans took a rationalistic approach to mathematics, considering integers to be the source from which all else flows"


"Pythagorean integer-mysticism, which heavily influenced Plato"

You are completely wrong in both, your characterization of Pythagorean metaphysics and the alleged "influence" on Plato.

Pythagorean metaphysics DID NOT stipulate "integers" as the ontological substance of Being, but "QUANTIFICATION". According to the Geometers, all of reality and Being are extensions and are therefore QUANTIFIABLE. Ofcourse, because they are quantifiable, it does not matter if the expressions of quantity are Rational numbers, integers, or Irrational numbers.

Secondly, Plato was definitely *NOT* influenced by Pythagorean metaphysics... atleast not positively. Plato - in many of his dialogues - rails AGAINST this quantifiable metaphysics and harshly criticizes the geometers. In the Platonic dialogues, Socrates explicitly argues against and laughs at the Pythagorean formulation that "man is the measure of all things"... ofcourse, knowing fully well that "measure" meant a literal usage of quantification.

In regards to your position that "validation" and "proof" is the same thing - atleast in Mathematics - I couldn't disagree with you more.
Mathematics is no different from a philosophical system so long as both paradigms use classical, symbolic logic in its methodology. In that respect, there are statements in Philosophy that can also be reduced to simple logical symbolism and stated identically to, say, a mathematical formulation.
In both cases then, you can have instances where certain statements will be valid but cannot be proven.
Reducing axiomatic statements to its symbolic expressions give you those instances where the statements cannot be proved.
For example (from Godel's book that I am currently reading), take the statement "all valid arguments are valid" - this is a tautology therefore it is valid, but it does not render itself to proof. You cannot prove that it is true... it remains self-evident due to the nature of the statement.
Now, reduce that statement to symbolic logic, and you'll get:
For any given x if P(x) and Q(x) then x has the property of Q.

2/14/2006 09:31:00 PM  
Blogger Adrian Hester said...

I don't think you actually know what you're talking about. First and most pitifully, "Man is the measure of all things" was said by the Sophist Protagoras of Abdera, not Pythagoras.

Second, the Pythagoreans (or at least many of them) did indeed consider everything material to start with number. That they considered quantity to be more basic is irrelevant to my point. As Aristotle wrote in his Metaphysics, "The Pythagoreans say that there is but one number, the mathematical, but things of sense are not separated from this, for they are composed of it; indeed, they construct the whole heaven out of numbers, but not out of unit numbers, for they assume that the unities have quantity; but how the first unity was so constituted as to have quantity, they seem at a loss to say." A consequence of this view among the earlier Pythagoreans was that all measures are commensurable, and thus that the world was built out of the integers; the proof that the square root of two is irrational knocked this belief into a cocked hat, hence the legend that Pythagoras sentenced its discoverer, Hippasus, to death by drowning.

Third, Plato certainly was influenced by the Pythagoreans. Again quoting Aristotle, "And Plato only changed the name, for the Pythagoreans say that things exist by imitation of numbers, but Plato, by sharing the nature of numbers." And further, "But that the one is the real essence of things, and not something else with unity as an attribute, he affirms, agreeing with the Pythagoreans; and in harmony with them he affirms that numbers are the principles of being for other things." All this is very well known.

2/15/2006 03:04:00 PM  
Blogger Adrian Hester said...

You write, "In regards to your position that "validation" and "proof" is the same thing - atleast in Mathematics - I couldn't disagree with you more.
Mathematics is no different from a philosophical system so long as both paradigms use classical, symbolic logic in its methodology."

Nonsense. Philosophy is centrally concerned with induction; mathematics proper is purely deductive, and any induction involved is outside the scope of mathematics--induction provides the statements you want to prove and the axioms you accept, but it is not used in proving anything. Note that classical symbolic logic does not allow for valid inductions--therefore, it must be different from a philosophical system, and so must mathematics (since you take classical symbolic logic as the sine qua non of mathematics).

"In both cases then, you can have instances where certain statements will be valid but cannot be proven.
Reducing axiomatic statements to its symbolic expressions give you those instances where the statements cannot be proved."

In other words, you didn't even understand the point of what I was saying! In mathematics, axioms aren't proved, they are not something to be validated in the system itself, they are irreducible primaries. Mathematics is concerned with proving by valid deduction the consequences of those axioms, and in that view, validity equals proof. (Which shows one of the limits of mathematics as a tool for the search for truth.) The validity of the axioms is a matter outside of mathematics. They are not a matter for proof within the system and thus the question of their validity is not mathematical and not a concern of mathematicians. (Of philosophers and mathematical scientists, yes, certainly it's a central question. You might think that mathematicians should be concerned with that, but they aren't, and if you claim that they should be then you simply misunderstand the whole thrust of contemporary mathematics, whether for good or for ill.) For example, if you apply mathematics to the physical world, the axioms (even if stated in pure mathematics) are chosen by non-mathematical criteria (abstraction from the properties of the part of the world being studied) and their application must be validated by experiment. However, in mathematics proper, that is irrelevant, since it is the system of relationships built on their basis that is important. This is precisely why you are dead wrong when you say that mathematics is just the same as a philosophical system--it is not, any more than any other science is, with the difference that other sciences rely on induction in their methodology. (This is one reason many mathematicians argue that mathematics is not a science.)

2/15/2006 03:28:00 PM  
Blogger Ergo Sum said...

It seems like your narrow-vision goggles simply do not allow you to grasp the entirety of any argument beyond your little insipid points.
Protagoras, Theodorus, Theaetetus were all part of the SAME SCHOOL of mystical doctrinaire that Pythagoras was part of! They were all geometers who shared the SAME METAPHYSICAL philosophy formulated by PYTHAGORAS that man is the measure of all things.
Plato explicitly argues AGAINST that philosophy - the fact that his eventual metaphysic of Idealism shared some fundamental similarities with the geometers is not what I am arguing against.

For the sake of your lousy attempt at argumentation, you pick my sentence out of its context and base your entire refutation on that sentence. I said:
"Mathematics is no different from a philosophical system SO LONG AS both paradigms use classical, SYMBOLIC LOGIC in its methodology."

Your entire argument is saying the same thing I said, except you also expend tremendous effort at trying to manipulate what I said into what you WISH I had said.
It is a ridiculous point to say that philosophy is ALL its systems is the same as mathematics because I am well aware that philosophy is not pure deduction. Hence, MY VERY EXPLICIT QUALIFICATION: Philosophy is the same as mathematics SO LONG AS BOTH are functioning in the paradigm of symbolic logic! Infact, it is BEYOND ARGUMENTATION that "Logic" was not even represented academically in a University's Mathematics department but was always a part of the Philosophy department. It was only recently (around 1940s) that Mathematics dept. in Universities began recruiting professors of LOGIC (or logicians) to be represented.
It goes to show that so far as both fields are functioning with interchangeable symbols arranged in a system of rules - both fields are the same.

I'm not interested in such petty and juvenile arguments with someone who doesn't even have the sense to read carefully the points he is attempting to refute. Get out of this blog.... I'll leave your posts on here, but if you post another thing anywhere here, I will delete everything that you've expended so much effort at writing. Have atleast the decency to have your own blog, and there you can engage in plentiful sophistry!

2/15/2006 05:37:00 PM  

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