Philosophy of Mathematics

1:30 p.m.  Peirce, Zeno, Achilles, and the Tortoise, 983-S1-351
 Daniel C Sloughter*, Furman University

2:00 p.m.  Structuralist Mathematics and MathematicalUnderstanding, 983-S1-544
 Kenneth Manders*, University of Pittsburgh

2:30 p.m.   Are Mathematical Objects Inside or Outside a HumanMind?983-S1-546
 Roger A. Simons*, Rhode Island College

3:00 p.m. What is Mathematics I: The Question, 983-S1-341
 Bonnie Gold*, Monmouth University

3:30 p.m.   A Conjecture about... Feminist Mathematics?  983-S1-429
 Sarah-marie Belcastro*, Xavier University

4:00 p.m.   Defining Mathematical Esthetics within the NCTMStandards, 983-S1-137
 Michael J. Bossé*, Indiana University of Pennsylvania

4:30 p.m.   Unfair Gambles in Probability (Preliminary Report), 983-S1-482
 John E Beam*, University of Wisconsin-Oshkosh

5:00 p.m. The Pedagogical Challenges of One to One Correspondence, 983-S1-551
 Satish C Bhatnagar*, University of Nevada-Las Vegas

5:30 p.m.   A Unifying Principle Describing How MathematicalKnowledge Unfolds, 983-S1-545
 M Anne Dow*, Maharishi University of Management

[Note:  The general theme of my talk is that the axioms from one theoretical area of mathematics could lead to meaningless results, not only in the "real world," but even in other theoretical areas.  Probability provides an interesting backdrop, because this subject is regarded by most mathematicians as a (semi-applied) sub-discipline of measure theory.  In fact, probability was not generally held to be "legitimate" mathematics until Kolmogorov placed it on the foundations of measure theory.  De Finetti believed these to be the wrong foundations.  He provided many illustrations of this; I will describe some of these and conclude with my example of an unfair bet.  I will not assume the audience to have any formal knowledge of measure theory or probability.]

In the 1930's, Kolmogorov borrowed the axiomatic system of the Lebesgue measure as a foundation for what is now the standard theory of probability.  De Finetti argued that many of the modern analytic developments are devoid of meaning in the context of probability.  In particular, he believed the assumption of countable additivity to be unjustified.  He proposed a broader alternative notion of a "coherent" probability, consistent with the Lebesgue theory, but requiring neither countable additivity of the measure nor any sort of structure on its domain.  A coherent probability can be interpreted as an assignment of fair odds for a bet.  I shall demonstrate that if one attempts to use an analogous notion to include the axiom of countable additivity, grossly unfair bets may result.

To illustrate, I will consider the Lebesgue measure P on the unit interval.  A "payoff function for a bet" is a function, the sum of a countable number of terms of the form  a[I(A) ­ P(A)],  where  a  is "a" real number,  "A"  is a Borel set and  "I"  is the indicator function.  "P(A)"  is the cost of a $1-payoff bet on A.  I will let c > 0  be given, and by exploiting the conditional nature of the convergence of the alternating harmonic series, I will construct a payoff function which is everywhere greater than c.  The clever gambler can be assured of winning an arbitrary amount of money.  This example does not rely on typical contrivances; for instance, only a finite amount of money is required of either the gambler or the house.

Remark:  A number of notable mathematicians (including Lester Dubins and Bill Sudderth) subscribe to de Finetti's interpretation of a probability.  Recently, the theory has been expanded to include a theory of integration and laws of large numbers.

Here's an example of a simple bet:  A gambler pays the house P({Heads}) dollars (in our example, 50 cents) for the promise of a one-dollar payoff if the outcome of the toss is Heads (and 0 payoff if the outcome is Tails).  The payoff is described mathematically by

This is a real-valued function defined on the sample space.  For instance, applied to the outcome Heads, the function takes the value 0.5 (the gambler netted 50 cents, because he got his dollar payoff and paid 50 cents for it).  Applied to the outcome Tails, the function takes the value -0.5 (the gambler got no payoff but paid 50 cents for the opportunity).

An example of a more complex bet:  A gambler pays the house 3 P({Heads}) dollars for 3 one-dollar-payoff bets on {Heads}, and SELLS to the house a one-dollar-payoff bet on {Tails} for the price of P({Tails}).  The payoff function is

Here, if the outcome is Heads, the payoff is 3[1 - 0.5] + (-1)[0 - 0.5], or 2 dollars.  (The gambler nets 2 dollars.)
If the outcome is Tails, the payoff is 3[0 - 0.5] + (-1)[1 - 0.5], or -2 dollars.  (The gambler loses 2 dollars.)

This payoff function is a linear combination involving two simple bets. De Finetti demonstrated that a finitely-additive probability measure is equivalent to an assignment P of odds for which it is NOT possible for a gambler to arrange any finite number of simple bets such that the resulting payoff function would be everywhere-positive over the sample space.  (In other words, a clever gambler could not guarantee himself a win.)

A natural conjecture would be:  A countably-additive probability measure is equivalent to an assignment of odds for which it is not possible for a gambler to arrange any COUNTABLE number of simple bets such that the resulting payoff function would be everywhere-positive over the sample space.  This conjecture turns out to be false, and that is what my example will illustrate.

Not many probabilists, even, are aware of these sorts of examples.  Much of modern probability theory is dependent on the axiom of countable additivity.  So many standard results of the theory are meaningless, at least in the sense that probability should be something more than a branch of measure theory.

[To clarify some of the technical details.  A "sample space" in probability is the set of possible outcomes of some experiment. It could be any type of set -- for instance, to model a single coin toss, the sample space could be {Heads,Tails}. The indicator function "I" is the function which is 1 on outcomes in the set, 0 on other outcomes -- in the coin-tossing example, "I({Heads})" would take the value 1 if the outcome of the toss was Heads, and 0 if the outcome was Tails. The probability "P" is a real-valued function defined on some collection of subsets of the sample space -- for example, a fair coin would be modeled by  P({Heads})=1/2, P({Tails})=1/2, P({Heads,Tails})=1.]

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At Maharishi University of Management, we seek fundamental principles unifying various branches of mathematics in order to help students appreciate how the topics they are studying relate to the whole discipline, to themselves, and to knowledge in general. This means that we are concerned with general principles governing the way in which mathematical knowledge unfolds.

The founder of Maharishi University of Management, Maharishi Mahesh Yogi, has stated that the fundamental dynamics of consciousness governing the entire universe, expressed in a key verse of the Vedic literature, are also necessarily expressed in the fundamental theories of every discipline.

In the mathematics department at Maharishi University of Management, we have located these dynamics in the major branches of mathematics: set theory, logic, the theory of the continuum, algebra, analysis, topology, category theory, and many others.

The key verse comes from the Rig-Veda:
"The verses of the Veda exist in the collapse of fullness in the transcendental field, in which reside all the laws of nature responsible for the whole manifest universe. He whose awareness is not open to this field, what can the verses accomplish for him? Those who know this level of reality are established in evenness, wholeness of life." [Rig-Veda I.164.39]

What does this mean? The idea of consciousness is fundamental to Maharishi's interpretation of this verse. Up to now the term "consciousness" has been excluded from scientific discussion largely because its meaning has been too vague and indefinite. Psychology has mainly dealt with isolated aspects of conscious experience. Maharishi, in his Vedic Science, has provided a highly coherent theoretical account of what consciousness is and a reliable, systematic method by which it can be isolated and directly experienced in its most fundamental state. In this account, consciousness is primary, not an emergent property of matter that comes into existence through the functioning of the human nervous system. It is a vast, unbounded, eternal, unified field, which gives rise to and pervades all manifest phenomena including the human mind, nervous system, and behavior. The method of experiencing it is the practice of Transcendental Meditation, which allows the mind to be drawn beyond being conscious of perceptions, thoughts, feelings, and even individual identity, to identify itself with this field, a state in which consciousness alone is.

According to Maharishi, "fullness" in this verse refers to the unbounded, all-pervading nature of the unified field of consciousness. By virtue of being conscious, this field eternally experiences itself as knower (of itself), as the process of knowing, and as known (the object of its knowing). "Collapse" refers to the flow of attention (knowing) from itself as "fullness" to itself as a single object of knowing. The "transcendental field" in which this collapse takes place is therefore the field of consciousness itself. Further interactions of the three: knower, knowing, and known, each of which is none other than consciousness itself, are reverberations within consciousness. These reverberations were experienced by the ancient seers of the Vedic tradition as a sequence of sounds within the deep silence of their minds, which were recorded as the "verses of the Veda". These reverberations of consciousness constitute the deepest aspect of the innumerable laws of nature, those which govern the eternal, unchanging, unified field of consciousness, and those which science uncovers in the phenomenal world. The final two sentences of the verse tell us that knowledge and direct experience of the unified field of consciousness are necessary for success and fulfillment in life.

Where is this pattern found in mathematics? Let's take, for example, the following view of the theory of the continuum. The theory of the continuum unfolds as a sequence of definitions, theorems, and proofs. These may be perceived as analogous to the "verses of the Veda".  Veda means pure knowledge, pure in the sense of consciousness knowing only itself. And these definitions, theorems, and proofs constitute knowledge of a slightly less abstract kind. The theory is based on ("exists in") the quantification of the continuum by the real numbers. This quantification takes place by means of nested infinite sequences of intervals, whose intersections are single points. Each of the sequences of nested intervals gives rise to an infinite decimal expansion that represents the point. Thus we identify the continuum itself with "fullness" and its quantification into points with "collapse".  The phrase "in the transcendental field" may be taken to correspond to the fact that this quantification is properly described and supported by the abstract field of set theory. Set theory permits one to define and manipulate the infinite sequences of intervals, and to collect the resulting infinite decimals together into an uncountable set with an algebraic structure and a natural ordering. Then "in which reside all the laws of nature" can be taken to mean, in this context, that set theory enables us to deal with the rules ("natural laws") governing the diverse structures of the continuum: its algebraic structure of addition and multiplication and its geometric structure based on the natural ordering and the topological completeness of the real numbers. If one then thinks of applications of the theory of the continuum, the phrase "responsible for the whole manifest universe" corresponds to the fact that the integration of algebraic operations and geometric continuity in the continuum of numbers makes it possible to represent and completely quantify any continuous process using transformations (functions) of the continuum within itself.

The remaining two sentences of the verse are clear if one considers the history of calculus in the 19th century. At the beginning of the 19th century the foundations of calculus were very shaky. For example, Fourier's seminal paper on heat propagation was rejected because of the limited understanding of the concept of a function and because of basic unanswered questions about convergence of series of functions. By classifying and formalizing the concept of the infinite, set theory provided a foundation for the theory of the continuum and hence for modern analysis.

There are many other examples of this principle to be found in the theories of mathematics. Students express appreciation for the unity of mathematical thought when they see this same fundamental principle operating in the different areas of mathematics they study.

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