Category Archives: Mathematics

Does Pi Exist?

To almost everyone this would seem a ridiculous question. Even to most mathematicians the existence of Pi (defined here as the ratio of a circle’s circumference to it’s diameter) is taken for granted. But the question of the existence of the class of number to which Pi belongs is a key example of a fundamental question in the philosophical understanding of Mathematics, and indeed of a broader issue in metaphysics and epistemology.

To understand the meaning of Pi, let’s start with a quick review of the types of numbers we encounter in elementary mathematics. We start with the “natural numbers” – the numbers 1,2,3 and so on. The very fact that these are the numbers we first learn and first teach to our children in their infancy is ultimately the key to understanding the issue in front of us. In the natural numbers, there are no negative numbers, and no zero.  The extension of the number system to include negative numbers I’ve discussed previously.  The concept of zero I’ll defer yet again, though I’ll hint that I believe it represents balance. 

 Both natural numbers and integers are “countable” – meaning that they can be ordered and indicated in order.  Considered as sets, they both contain a “countable infinity” of elements. 

The next extension of the number system is to include all fractions – the division of natural numbers by natural numbers, and while we’re at it we throw in division of signed integers by signed integers other than zero.  This new number system we designate as the “rational numbers”.  The natural number or the signed integer “n” is represented in the rational number system as ratios n/1, 2n/2, 3n/3, etc.  (Note that I say the signed integers are “represented” – I maintain that the signed integer n is different than the rational number n/1, 2n/2, …).   The nature in which rational numbers exist is again the topic for another discussion, though I’ll note that their nature is related to the epistemological process of measurement, and the mathematical concept of continuity.  I’ll also note that the rational numbers can be so ordered as to be countable, and so are “countably infinite” in number.

The next step is a bit tougher to follow.  The “algebraic numbers” are defined as all numbers which are solutions to polynomial equations whose coefficients and exponents are natural numbers.  In a simpler form, it is this number system that sparked the first significant philosophical debate about the meaning of numbers – back in Greece in the time of Pythagoras.  What we now refer to as the “Pythagorean Theorem” states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other sides, or a^2+b^2=c^2, where c is the hypotenuse.  In its simplest form, if we are computing the length of the diagonal of a square with unit length sides (a=b=1), then c*c=2.  That the value of c, the square root of 2, cannot be expressed as a fraction of natural numbers was a major discovery of Greek mathematics. (This fact was proven by members of the Pythagorean school, possibly by one Hippasus who may have been executed for revealing the proof because it had been held as a state secret).  A good deal of debate occurred through the centuries over the existence of these “irrational numbers”.  I have not spent enough time considering the nature of these numbers to offer an opinion as to what they represent, but I will venture that this number system will be found to have an acceptable basis in reality.

As with the rational numbers, the algebraic numbers can be ordered in such a manner as to be countable.  There are therefore a countable infinity of algebraic numbers.

Both the rational and algebraic numbers can be shown to be “dense” on the real number line.  Think of the real number line as an “infinitely long” straight line, as you did in high school.  If we take any interval of this line, there will be contained in that interval an infinite number of rational numbers, no matter how small an interval is chosen.  This is the density property of a set of numbers.  This observation, in combination with the fact that all of these number classes are “countably infinite” leads to some confusing conclusions.  In the sense that we can order and count the elements of these number classes, we can establish a “one to one” relationship between all rational numbers with the natural numbers.  “In a sense”, this implies there are “as many” rational numbers as natural numbers – despite the fact that between any two natural numbers there are an infinite number of rational numbers!  The resolution of this confusion lies in the use made here of the concept “infinite” – which ultimately is not a valid concept.  But that is another digression.

The last development in this trail of number classes is the discovery of transcendental numbers.  The historical development of this number class derived from the problem of “squaring the circle”, which was a phrase used to describe the problem of computing the area of a circle with a rational-valued diameter in terms of algebraic numbers.  It was shown in 1882 that this is impossible, because Pi is not the solution of any polynomial with natural number coefficients.  Pi then belongs to yet another class of number.  Rather than deal with the area problem, let us define Pi in the simplest possible manner.  Pi is the ratio of the circumference of a circle to its diameter.  Hence, if a circle has a diameter equal to a natural number of units (or a rational, or even algebraic number of units), its  circumference will be a transcendental number. 

Using this definition, we have for the first time a number tied to a perfect geometrical figure.  A circle is an ideal figure that can never be found in the concrete world of objects.  Any “circle” you encounter in the physical world will be imperfect at some level.  “Circle” is a idealization of these experienced real-world circles.  It can be thought of as the limiting perfection of all physical circles.  And so, the question of whether Pi “exists” will become the question of whether this idealization can be said to exist.  More generally, it is the question as to whether a conceptual idealized abstraction can be said to exist.  Such an abstraction can never be exhibited in the physical world.

The answer to this question lies in the fundamental philosophical principles that are applied.  The empiricist will require that to “exist”, a thing must be present in the physical world, and furthermore, must be experienced.  Such an approach will then conclude that ideal circles do not exist, and therefore Pi does not exist.  The rationalist/Platonist will be happy to place the idealizations in their own realm (ultimately a realm occupied by a God), and declare that they in fact exist in this realm, but can only be approximated in the denigrated physical realm.

Finally, the Objectivist will answer with clarity.  The concept circle – as with all other human concepts – exists as a relationship between the human mind and the physical world.  To exist, the concept must have physical representations from which the details specific to each (such as imperfections) have been abstracted away (omitted as measurements) to arrive at the concept in human consciousness.   The concept circle then exists as an epistemological linkage between human understanding of the physical world, and the real world itself.  And since the circle exists, it is in the same sense that the number Pi (and ultimately all transcendental numbers) exist.

The acceptance of transcendental numbers leads to additional confusions involving the invalid concept of infinity.  Unlike the other number classes, it can be proven that there is an uncountable infinity of transcendental numbers.  A one to one relationship between the transcendental numbers and the natural numbers cannot be established.  Indeed between any two rational numbers (or even algebraic numbers) there is an uncountable infinity of transcendental numbers.  (Intriguingly, a one-to-one relationship can be established between transcendental numbers and the set of all subsets of natural numbers, which in turn implies that the set of all subsets of natural numbers is uncountably infinite as well). 

The further discussion of how these “infinities” and the concept of continuity and “infinite divisibility” should be approached to allow a resolution of the apparent contradictions will need to wait for another time.  Interested parties should consult writings and lectures given by Pat Corvini for excellent discussions of these topics.

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On Probability

Over this winter vacation, I’ve been intellectually focused on some background mathematics supporting my work in detection. One of my projects is reading through a rather introductory text on probability by Athanasios Papoulis (1965). I’ve found the opening sections of this book very philosophically relevant, in addition to being a promising text book – if I can ignore the philosophy being presented.

Papoulis opens the text with a very blunt statement, which instantly alerts me to the conceptual framework in which this mind is operating: “Scientific theories deal with concepts, never with reality”. What follows in his introduction is a justification for approaching the subject purely from a deductive process starting with a set of axioms, and not worrying – much – about the relationship between the theory and “the real world, whatever that means”. Among the blunt statements in the introduction is this:

To conclude, we repeat that the probability P(A) of an event A must be interpreted as a number assigned to this event, as mass is assigned to a body. In the development of the theory, one should not worry about the “physical meaning” of P(A). This is what is done in all theories.

After this very open confession of intellectual sterility, the author gets down to work by asking how probability – which he has just declared to be arbitrarily defined numbers – should be defined. He offers three strawmen definitions, and then settles on a fourth. The strawmen are of some interest as well:

(1) Relative Frequency Definition. This is probably the definition most commonly assumed – that probability is the ratio of the number of outcomes of interest, divided by the total number of trials. This needs to be more carefully constructed as the limit of this ratio as the number of trials “goes to infinity”, or, as I’ll even more carefully say it, as the number of trials becomes arbitrarily large. Papoulis rejects this approach as too cumbersome, because no matter how many actual trials are made, we never approach “infinity”, and therefore can never say that we have a sufficient number of trials to establish meaning for this ratio.

(2) Classical Definition. Here the probability of an event is determined by a-priori reasoning about the situation at hand. For example (Papoulis’ example) a six-sided (fair) die has six equally possible outcomes, so the probability of getting a one on a die roll is 1/6. No experiment need be made to reach this (rational) conclusion. This strawman is knocked down by stating that this really only can work for simple cases, and works most easily only when the outcomes are of equal likelihood. A couple examples are given to show that determining the proper probability in this manner is very error prone. For example, one could attempt to determine the likelihood of rolling a 2 with two die by saying the number of possible outcomes is 11 (2,3,…,12), and we are interested in the value 2, so the probability is 1/11, which is of course wrong.

(3) Measure of Belief. This one seems to be thrown in as an easily dismissed psychologically based argument.

Papoulis chooses to define probability from three axioms, and claims to then proceed to develop the entire theory of probability from these axioms (plus one more minor extension). The axioms seem ridiculously primitive:

The probability of an event A is a number P(A) assigned to this event. This number obeys the following three postulates, but is otherwise unspecified:
I. P(A) is positive or zero
II. The probability of the certain event is 1 [the certain event always occurs]
III. If A and B are mutually exclusive [they both cannot happen in the same trial] then P(A+B)=P(A)+P(B). [The probability that A or B happens is equal to the sum of the probability that A happens and the probability that B happens].

And that’s it! That’s his “definition” for probability, which clearly lacks any meaningful tie to the “real world, whatever that means”. So he can proceed in philosophical comfort.

Or can he?

Interestingly, in order to start his progression from these axioms, he needs to introduce a large segment of set theory – otherwise he cannot define what an “event” is, which is contained in this definition of probability. Without belaboring set theory here, an “outcome” is one possible result of a trial of the process we are trying to test. The set of all possible outcomes is the “certain event” mentioned in the definition – one element of the certain event will always be the outcome of any trial, so the probability of the certain event is 1. An “event” is then any subset of the set of all possible outcomes.

Next, we encounter a very bizarre twist in this “axiomatic” probability theory. Not all events, says Papoulis, can have a probability assigned to them. That is, not all sets of possible outcomes can be given a number that will meet the axiomatic conditions from which the theory of probability will be developed. Papoulis does not confront this problem directly, treating it as merely an annoyance, and refering the reader to measure theory for a better explanation, but he does give a major example to indicate where the problem lies. Suppose the outcome of a process that we are interested in could be any real number (real numbers are all of the common numbers from “negative infinity” to “postive infinity”, including all rational and irrational numbers). Then the “certain event” is the set of all real numbers. But consider the event which is the set consisting of a single number, say the set {3}. If every set of this type is given a probability, there will be an “infinite” number of these probabilities, and in order for axiom III to remain true, each of these probabilities would need to be zero. Then for any specific outcome of a trial of the process – A -, the event {A} would have a probability zero – which is a clear contradiction.

The escape from this contradiction is itself quite bizarre, and only partially explained (we are referred to measure theory for a more complete explanation). For this example, only events that can be formed from the union or intersection of a countable number of continuous intervals or isolated points will be given a probability. There are sets that cannot be so formed (we are told they are complicated to construct, and I recall similar constructions from my days of formal math training, and I agree with him), and these will not be given probabilities.

If this escape seems to make little sense – and Papoulis seems to understand that the reader will not be able to make sense of this – he offers a better escape: “…one can construct certain pathological sets that are not countable intersections or unions of intervals. Sets of this kind have no probabilities, but are of no importance in applications, and we can forget them“.

Now this is a simply amazing demonstration of a rationalist getting boxed into a corner, and then escaping inelegantly by refering us back to the real world (whatever that means) which he has already denied can be consulted in developing a proper mathematical “theory”.

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Philosophy of Mathematics: Randomness, Part I

One of the most fundamental principles within mathematics is randomness. The study of random processes is the direct subject matter of probability and statistics, from which a broad assortment of additional fields arise, including game theory, and the theory of stochastic processes. Applied mathematics is heavily populated with applications of statistics, in finance – from market dynamics to insurance models, in physics – from thermodynamics to quantum mechanics, and throughout engineering in the form of error budgeting, tolerancing, safety factoring, and risk analysis.

What, precisely, is meant by randomness? A common definition is that a process is random if no order can be discerned in its manifestation. A more mathematical definition may include the necessity of unpredictability, a lack of bias or correlation to other processes, and may require that though “random”, the process output must follow a particular probability distribution. In a higher-order of randomness, which has been referred to as an “arbitrary” process, no probability distribution in particular applies to the process output. A more philosophically-oriented description may assume a-causality and non-determinism in the events depicted as “random”.

In the quest for the proper philosophical foundations for the fields of mathematics arising from the concept of randomness, two main questions require consideration:

1)What form of randomness exists in Reality?
2)What is the relationship between the philosophical understanding of randomness, and the nature of randomness required by the fields of probability and statistics?

It is an axiomatic, self-evident truth that all physical events are caused by prior events. The Law of Causality therefore forbids the existence of random physical processes exhibiting a-causal and non-determined behavior. On the other hand, there can exist causal, but unpredictable behavior.

Unpredictability can take two forms. What I will term “soft” unpredictability arises from a lack of knowledge on the part of the entity attempting a prediction. Soft unpredictability is not inherent in a process, but lies in the current state of the predictor. By educating the predictor, the unpredictability can be removed. Hence, I cannot predict my company’s cafeteria’s lunch menu for tomorrow only because I am not in communication with the manager of the cafeteria service. My ability to perform this prediction could be altered with an appropriate phone call (if I knew the name or number of the cafeteria manager).

By prediction, I mean specifically the ability to indicate the future state of a process at a time prior to the time at which the process will achieve that state (the prediction must also be for a particular future time). This is an essential distinction to consider in the description of what I term “hard” unpredictability.

There is a vast category of phenomena in which infinitesimal changes in starting conditions lead to exponentially divergent outcomes as the process evolves. This is the (approximate) mathematical definition of a chaotic process. For purposes of illustration, consider a simple mathematical series: x’=Gx(1-x), with G=3.7. On each iteration, take x=x’ and repeat the calculation. If we run two series A and B, with the first value in A being 0.1, and the first value in B being 0.100001, we get the following results:

Start 0.1 0.100001
20 iterations0.256724484 0.256693475
30 iterations0.682868078 0.68171437
40 iterations0.767514785 0.750851267
50 iterations0.608389182 0.924135972

Beyond this point, the two series are completely uncorrelated (as will be very obvious if you plot the series). In this example, an accurate prediction of the 50th iteration of the series requires knowledge of the starting point to better than 1 part in 100,000. As the series progresses, the maximum allowable error of knowledge in the starting condition to allow accurate prediction decreases exponentially. This characteristic of chaos is what I am terming “hard unpredictability”.

(There is much, much more that can be said about our example, which is known as the “logistic series”. For the adventurous: run various examples varying the value of A from 0 to 4 to see a variety of behaviors, particularly right around A=3.57. If you then think you’ve got a handle on it, try A=3.82.)
– To be continued

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Notes on recent reading

I interrupted my chain of thought on the philosophy of mathematics by reading some published material on that topic. I had “told myself” that I should avoid doing this, in the interest of keeping my thoughts fresh and free of any confusion that could come from reading other possible interpretations which may be based on unsound principles. Nonetheless, I proceeded to order a few books through the public library, based only upon titles, and ignorant of who the authors were.

I wound up with three books, of which I bothered reading only one – The Philosophy of Mathematics, by Stephan Korner, published in 1960. I’ve read a string of books now which all suffer from a common problem – incredibly poor writing style – and this one just added to the sequence. Although it is possible that I am finding the subject just too complicated to understand, Korner’s writing seems a torture of partially and poorly defined words and phrases. I’ll also admit to reading this book far too quickly, not taking notes, and being deeply bored by the experience. Through this haze, he presents three major schools of thought on the nature of Mathematics: Logicism, Formal Systems Theory, and Intuitionism.

The Logicists (Frege, Russell, Whitehead, Quine) seek to derive Mathematics from the fundamental rules of Logic. This school is a direct derivative of the rebirth of rationalism after Kant. Mathematics is a product solely of the mind, and bears no relationship to Reality. Mathematical truth is a synthetic a-priori truth: existing in a realm beyond experience, but created by the human mind. Not surprisingly, this attempt is seen to fail.

Formal systems theory is the attempt to consider Mathematics as a language of symbols, and to use this language representation and a set of transformation rules (the rules of Logic) to attempt to show the completeness or incompleteness of a set of axioms. This approach culminates in Godel’s Incompleteness Theorem, which shows, for the specific field of Number Theory, that given any assumed set of axioms, one can construct a “true” statement relative to those axioms which cannot be proven starting from those axioms and using the transformation rules. This esoteric proof is then generalized (foggily) to apply to all of Mathematics, and indeed, to all of Logic. Basically, this approach to understanding Mathematics results in the conclusion that mathematical truth is by its nature incomplete, and therefore of questionable application.

Korner’s exposition of Intuitionism I found to be almost entirely opaque. This approach is strongly empirical, and requires of every truth that it be inherently provable, and constructable. This leads to the denial of the law of the excluded middle: A or not A, as to prove the existence of a thing, it is not sufficient, according to the Intuitionist, to prove the impossibility of its non-existence. All existence proofs are required to lead to a construction of an example. I cannot claim to understand intuitionism – once the excluded middle was denied, I decided to spend my time elsewhere.

All three of the schools Korner describes are derived from Kant’s disastrous epistemology, as Korner correctly indicates. The author describes each school, and then offers his criticism, ending the work by proposing some philosophical framework of his own. My interest diminished rapidly after the introductory chapters, in which he discusses Kant with obvious reverence.

The other books I picked up included a set of lectures on mathematics by Wittenstein – after reading up on Wittenstein a bit, and discovering he was a self-contradicting linguist, I abandoned all hope of getting anything out of his impossible to read lectures.

The final book is more modern (1990s), entitled “What is Mathematics Really?” by Reuben Hersh. I have not opened this book, but Hersh apparently presents his “humanism” theory of mathematics, in which Mathematics is primarily a social activity which resists a firm definition. “Mathematics is what Mathematicians do.” How insightful. Reviews indicate that the presentation is very poorly constructed, that the history he presents is inaccurate, and his arguments range in quality from silly to questionable. I believe I’ll return this one to the library unopened.

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Multiplication

As another example of the possibility of confusing number classes, consider the difference between 5 apple trees containing 100 apples each, and 100 apple trees containing 5 apples each. The equality of the number of apples in each of these cases is asserted in the abstract expression 5×100=100×5. Although when considering soley the number of apples, this equality is certainly valid, it can become misleading to consider these two expressions “equivalent” in any sense more specific than that of the abstract counting of apples.

The two terms of a multiplication are formally labeled the multiplicand – the number of items to be multiplied – and the multiplier – the number of times the multiplicand is to be repeated. This distinction is critical in understanding the process of multiplication, and the proper meaning of the multiplication. Again the issue is in the class or type of numbers being combined. Here I will restrict the discussion to multiplication of positive, real numbers by natural numbers. The multiplicand is either a count of units, or a measure of a continuous quantity in units. The multiplier is a natural abstract number representing a (whole) number of repetitions. When separated, these numbers are not of the same class, and therefore cannot be interchanged in two expressions and considered equivalent. When considered in this manner, 5 sets of 100 units each is not equivalent to 100 sets of 5 units each. Only when the multiplication is “computed”, and the resulting total measure taken, can we state that the two products yield an equal measure of units.

The danger in confusing the two levels of meaning present in a simple multiplication, such as 5×100 and 100×5 becomes apparent when abstract substitutions are made, out of the context of the computation of units, in a lengthy computation. Unless the mathematician is taking great care to interpret the meaning of his expressions, a valid sequence of abstract mathematical expressions can lead to a meaningless result.

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Distinctions between Number Classes

In the discussion of signed integers, it was noted that the nature of arithmetic involved in their use was fundamentally different than arithmetic performed with natural numbers. This fact is a result of a deeper distinction between these numbering systems. A natural number is not a signed integer. The correspondance between positive integers and natural numbers is one of analogy, but not of identification or definition.

It is perhaps easier to understand the distinction between classes of numbersby considering the historical development of arithmetic and geometry. The oldest examples of arithmetic consist almost purely of addition, subtraction and multiplication. When the need to divide arose in real-world problems, the results were expressed (in Sumerian, Babylonian and Egyptian “texts”) as combinations of natural numbers and sums of ratios of natural numbers. Thus an answer would beÂexpressed as 3 + 1/6 + 3/8. In fact, in ancient Egypt a sophisticated system of reducing such expressions to a standard form N+1/a+1/b+1/c+… was devised to “simplify” this usage. It is to be noted here that the idea ofbeing able to “combine” the various fractions and natural numbers was missing from these earliest arithmetic concepts – the math was much more cumbersome, but more conceptually honest at the dawn of our history.

Thedevelopment ofgeometry wasan entirely separated form of study in ancient history. Geometric problems were worked out usingconstructions based upon assumed self-evident axioms. The use of numbers to represent geometric concepts is first seen in classical Greece, but did not fully reach its maturity until as recently as the time of Descartes. A principle concern in geometry is the concept of “length” of line segments, or “distance” in general. The measurement of length using an established unit leads immediately to the need to express partial units, as length is a continuous attribute, unlike the count of a number of discrete items. In this key extension of the meaning of “number” from counting to measuring continuous value – and in particular the length of a line – a new class of number was introduced, encompassing the concept of integral subdivisions of units, or fractions.

The Greeks (first Pythagoras, or perhaps one of his students) first noted the incompleteness of this new numbering system, when considering the length of the diagonal of a square with a side one unit in length. It is a fairly easy proof (when using today’s notation) to show that this length is not a fractional expression of the side length. Hence, there must exist additional “numbers” that arenot expressible as fractions. It was over 1000 years later when Dedekind completed the identification of this new numbering system with the “real numbers” that represent- and can be represented as – lengths on a line.

The point to be made here again, is that the natural number system is fundamentally different from the fractional numbering system. That the fractions 8/4 and 2/1 are equivalent is a true statement; however that these numbers and the natural number 2 are equivalent is not at all true. As to whether the fractional numbering system is of the same class as the real numbering system is a more difficult question, depending on the conceptual framework from which the fractional numbering system is constructed. If, as presented here, the fractional numbering system is meant to represent the possible magnitudes of a value of continuous extent, then the fractional numbering system is incomplete, and must be extended to the (positive) real numbers to form a meaningful concept. [There is circularity in that statement, through the inexact use of the term “continuous”, which requires further thought]. On the other hand, if the fractional numbering system was developed as, say, a monetary reckoning system for trade in tangible items, it may be considered as a separate number class.

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Signed Integers

The natural numbers (the non-negative integers) are abstractions of the additive combinations of units. The number 10 is an abstraction of the set of all sets of that number of units. (This definition is similar to that given by Bertrand Russell).

In the consideration of the rational foundations of Mathematics, one of the first concepts which is a cause of concern is that of the “negative numbers”. As conventionally presented, negative numbers have no grounding in Reality. One can not have -3 cows in a field. There is either some natural number of cows present, or no cows at all. (I am deferring the concept of zero for a future discussion).

However, there are valid interpretations for the negative number notation. Consider the concept of electric charge, and assume, for the moment, the existence of unit quantities of “positive” and “negative” charge. To measure the “total charge” of a body, we count the number of positive charge units and subtract the number of negative charge units. Invoking the concept of negative numbers, we then obtain the total as a positive or negative integer. The proper interpretation of this total is the positive number of non-cancelling units of positive or negative charge. Note that the number of units in the total is always positive, and it is the nature of the unit which changes as the sign of the integral total is changed.

For clarity, lets try a second example. Determining the value of financial assets, we count the number of units of wealth, and subtract the number of units of debt. Both the number of wealth units and the number of debt units are positive values. These units cancel one another, and the total value is then the (positive) number of non-cancelling units. The sign of the total is set based upon whether the excess units are of wealth or debt.

The use of signed integers represents the combination of dissimilar units in arithmetic. The units represented must be of equivalent magnitude and exactly opposite meaning. This implies that any of the classes being combined in an arithmetic summation or subtraction, whose result will be a signed integer, must be subclasses of a class of entities measurable with a unit whose magnitude is common to the subclass units. The sign of the result is set based upon which of the opposing units is in excess within the total.

That was tough to write and tougher to read. In our first example, “charge” is the parent class, “positive charge” and “negative charge” the subclasses. A unit of “charge” is of the same magnitude as that of a unit of “positive charge” or “negative charge”. When I write C=P-N, with C=Total Units of Charge, P=Units of Positive Charge and N=Units of Negative Charge, I am combining equal magnitude but cancelling units of Positive and Negative Charge to obtain a total in units of the parent class “Charge”. The sign of this value is determined by which of the subclasses is in excess.

It is essential that the distinction between arithmetic involving natural numbers, and arithmetic involving signed integers be maintained in the use of mathematics. The underlying arithmetic meaning of:

Cows to Bring Home = Cows Brought to Market – Cows Sold

is not at all the same as the underlying arithmetic meaning of:

Total Asset Value = Wealth – Debt

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The Fundamental Axiom of Mathematics : The Unit

The fundamental axiom of mathematics is as simple to state as the fundamental axiom of Objectivist philosophy:

1=1 (Source: Ron Pisaturo)

What is meant by thissimple statement is that the unit measure of any class of entity, quality, or relationship is invariant across the domain of its measurement. The first apple in a barrel is the same (qua apple) as the 10th, 100th or 1000th apple. The distance measured by the first mile on the commute to work is the same as the distance measured by the 13th.

The concept of a unit,abstracted from that which the unit measures, is the concept which Mathematics labels as the number 1.

The concept of equality implies commensurability. By commensurable I mean that the referent which the symbol on the left side of the equation represents is of the same kind as the referent represented on the right side of the equation. Further, the precise definition of commensurate is “measured by a common standard”, here meaning measured by a common, universally constant, unit.

This needs slight elaboration. The “kind” which must be constant between sides of an equation is precisely the kind for which the unit is defined. This may or may not mean that the entities represented on both sides are of fully identical classes, but only that at the level of representation in the equation the classes are identical. Lets say we have 10 moose and 10 monkeys (because, as my kids know, I love talking about moose and monkeys). As moose and monkeys, these entities are not commensurable. The expression 10 Moose = 10 Monkeys is therefore not an equation, because the units do not match. However, in the abstract category “Mammals” to which both moose and monkey belong, we have an abstract unit “Mammal” which can measure both moose and monkey. The expression 10 Mammals = 10 Mammals is a valid equation, and in this context we can properly write 10Moose = 10 Monkeys and imply this equation.

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Philosophy of Mathematics – Introduction

Some recent reading has lead me back to considering the philosophical problems of Mathematics. To arrive at a clear and proper understanding of mathematical principles, it is necessaryto build all mathematical constructs from abstraction of entities occuring in reality. The historical introduction of many (perhaps most) mathematical devices and techniques has been driven by their pragmatic utility in addressing problems in the existing framework of mathematics. Failure to tie these new constructs to a basis in reality (physical reality, or abstraction fromphysical reality), results in an incomplete understanding of the meaning of these constructs. This can lead to inappropriate application of the constructs to model phenomena, which in turn can undermine the validity of theory based upon the arbitrary application of the mathematics. Even if thescience used in developing and verifyinga theoryuses a valid method,if the mathematics used to describe the theory is based in floating abstraction, the theory remains vulnerable to misinterpretation and misapplication.

My inspiration for pursuing an investigation toward properly rooting fundamental mathematics derives from a small number of sources. I have previously read through the final two of a series of articles (published in TIA in the late 90’s through 2001) by Ron Pisaturo in which criticisms similar to those I’ve described above are presented, with a handful of specific examples. Meanwhile, I am currently reading The Road to Reality by Roger Penrose, which is a lengthy text seeking to present the major elements of modern physics to the general reader through emphasizing the underlying mathematics. At the moment, I’ve completed only about 120 pages, but it is already clear that Penrose’s philosophy of mathematics is otherworldly – he describes a Platonic realm which contains the ideals of mathematics, and presents a rather confusing, but not revolutionary, set of relationships between the human mind, the physical world, and this Platonic realm. This alone was enough to indicate that Penrose is personally comfortable with working with floating abstractions, but his repeated description of the “magic” of complexnumbers makes this much clearer.

This is not to say that the book is without merit – in fact, for the purpose of understanding how the philosophical errors of modern physics stem from the philosophical errors of the underlying mathematics, the book may be ideal. And beyond this purpose, I chose the book(without knowing its content) to attempt to come closer to at least being conversant in elements of modern physics. [For those reading this without knowing me personally, I shouldmention that I have undergraduate degrees in both physics and mathematics, so these topics are not – or at least should not – be alien territory for me]. As to whether Penrose accomplishes his goal of making the mathematics of modern physics approachable for the general public, I highly, highly doubt that a reader without at least an engineering degree is going to be able to make it through the first 100 pages without becoming intensely confused. And these 100 pages are only scratching the surface of the math he covers. A full critique of Penrose’s book will come later.

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Thoughts on Number Theory

The conversation about life frequency has lead me on a short intellectual stroll. I’ve wound up reflecting on number theory, a subject with which I am only tangentally familiar, but one which holds me in a certain fascination. As we will see, this fascination is in turn of interest.

The notes on the January 12th meeting indicate my perceived connection between life frequency and cellular automata. My history of playing (off and on) with cellular automata goes back a very long way. In fact, it was the Game of Life distributed with the original Macintosh systems that first sparked my interest in cellular automata (no, I never owned a Mac, but the University of Rochester bought their first personal computer lab from Apple – I found those machines very painful and confusing to use, but that’s another story). In college, I spent a bit of time collecting information about these systems, and then, after college, I spent plenty of Perkin-Elmer’s time (unbeknowst to them), continuing this research. It was in the course of pursuing this interest that I first read Steven Wolfram’s papers on the topic, which is where he got his start in his theory of “a new kind of science” as he calls it. Subsequently (according to Wolfram), he decided he needed a more powerful system with which to explore his ideas, and he set out to build it. The result was Mathematica, which is by far my favorite software system. Wolfram then went on to publish his “New Kind of Science”, which he views as the completion of his quest for a unifying principle for complex systems of all kinds, including the Universe as a whole. In my opinion (this would be a separate topic), he falls far short of a theory of anything in this work, though he does make a slew (a big slew) of interesting observations.

Which leads me back to number theory (admittedly after Kim asked me one night what a Mersenne number is). Number theory also appears to consist of a huge pile of facts, which I’ll call observations, a somewhat smaller pile of recognized patterns within those observations (hypotheses), and a relatively small pile of proven relationships (theorems). The topic of number theory is (rather obviously) numbers, specifically positive integers; however, in advanced subfields of number theory this is generalized to include any “number system”. For our present purpose it is sufficient to consider just plain “elementary” number theory (which is far from elementary).

To get flavor for what elementary number theory contains, here is a very small taste of definitions, theorems, and assertions:

A Mersenne prime is a prime number of the form 2^n-1. The first few are 3,7,31,127,8191,131071,524287,2147483647. There are only 43 known Mersenne numbers. The largest one known has over 9 million digits. The 35th-43rd were “discovered” using a distributed processing computer system (GIMPS) that is downloaded and used similarly to the SETI screensaver.

Unproven hypothesis: There are an infinite number of Mersenne primes

A perfect number is equal to the sum of its divisors. The first couple are 6=1+2+3, 28=1+2+4+7+14.

Theorem: For every perfect number, there is a Mersenne prime, and vice-versa.

Unproven hypothesis: There are no odd perfect numbers. What is proven is that there are none less than 10^300 (1 followed by 300 zeros).

Theorem: Every even perfect number > 6, is of the form 1+9Tn, where Tn is a triangular number, which by definition is of the form 0.5n(n+1), with n=8j+2, j a positive integer.

Theorem: Every even perfect number is the sum of consecutive positive integers starting with 1 and ending with a Mersenne prime.

Theorem: The only perfect number of the form 1+x^3 is 28.

Had enough? To give you some idea of the breadth of number theory, there are definitions of each of the following types of numbers, each with its own theorems and inter-relationships to various other types:

Abundant Number, Amicable Numbers, Deficient Number,e-Perfect Number, Harmonic Number, Hyperperfect Number, Infinitary Perfect Number, Multiperfect Number, Multiplicative Perfect Number, Pluperfect Number, Pseudoperfect Number, Quasiperfect Number, Semiperfect Number, Smith Number, Sociable Numbers, Sublime Number, Super Unitary Perfect Number, Superperfect Number, Unitary Perfect Number, and, last but not least: Weird Numbers.

One more example: There are two Sublime Numbers known to man: 12, and
608655567023837898967037173424316962265783077335188597
0528324860512791691264.

We do not know if any odd sublime numbers exist.

If you are laughing by now (I was), ask yourself why. What is it about this field of math that seems ludicrous?

I’ll have more to say on this.

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