Monthly Archives: June 2006

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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Before Continuing…

I think thismay a good point to remind the reader of the purpose of this Blog, particularly since I have attracted some traffic other than my close friends and acquaintences. I use these pages to explore my thoughts on a variety of subjects. Up to this point, this has consisted mostly of commentary on other reading, or of history. In a sense, this replaces a journal I have kept for several years – the advantage here is the ease of typing over writing, and the ubiquitous availability.

What this body of material does not represent – necessarily – are finalized conclusions stated in a publication format. Most of what I have written here I have not seen a need to correct, but as these posts are written with very little editing, some of the discussion is poorly framed or otherwise awkward. The eventual goal is to later review the posts and extract material to form essays on particular topics of interest.

As I move into the next phase of posts, several of whichIhope will focus on the philosophy of Mathematics,I will include some original content, and I want to clearly indicate these caveats:

  • The ideas presented will be a product of my own thinking, but mixed with, and at times dependent upon, the thoughts and writings of others. I do not promise to fully cite each reference I am using, though some attempt to indicate the source of major ideas (where I remember the source) will be made.
  • When I contradict myself (in the process of developing this body of thought, this may be inevitable), I will call out the contradiction. If I do not, and you the reader find a contradiction, I will appreciate a comment to that effect.
  • I will nominally try to present this material in a logical sequence, although most likely mixed with other topics; however it is very likely that in the first attempt the presentation will not be linear.

Ok, with that out of the way, lets get on with the fun.

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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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