This article provides a philosophic development of the fundamental concepts, principles, and methods of mathematics. Since this article does not deal with advanced mathematics, you, the reader, will not learn the mathematics needed to build bridges or rockets. If, however, you ever do study the mathematics of rocket science, this article may help you to understand it, and any mathematics, much better. You will see mathematics as a system of principles and methods—principles induced from factual observation and methods devised to measure aspects of reality.
There is still another reason you may want to study this material. Since the reasoning used is largely inductive (as opposed to deductive), the material provides insight into the nature of inductive logic.
Let us develop the foundation of mathematics from the very beginning, assuming we know nothing about mathematics. We will start with the following situation that cavemen, who knew no mathematics, might have faced. Suppose we sent each of our children to a different location to fish, and each child returned with a pile of fish. To which location should we send all of our children tomorrow? All the fish are virtually the same size and almost identical in every perceivable way, but the piles are not the same size. Which pile of fish is the largest?
The Foundation of Mathematics (Part II)
by Ronald Pisaturo and Glenn Marcus
September 1994, Vol 8 No 5
Measurement Without Counting
Measurement With Counting: An Example
A More Advanced Example With Philosophical Explanation
Unit and Unit of Measure
The Logical Structure of Mathematics
The Number System
Summary and Conclusion
In Part I of this article, we traced the logical development of counting, the most basic method of mathematics. When we count units in a group, we are measuring the quantity of the group. Now we ask: what about measuring the size of something that is not a group, but a single entity? For instance, what if we want to measure the size of a pile of applesauce, the length of a spear, or the weight of a rock?
Mathematics in One Lesson (Part I)
by Ronald Pisaturo
September 1998, Vol 12 No 9
My colleague Glenn Marcus tells a story of when he was a professor of mathematics at a city university. At one faculty meeting, Glenn mentioned that it was important to make sure students understood what multiplication by negatives means. One professor replied that he did not really understand what multiplication by negatives means.
The prevailing schools of thought in the profession of mathematics have fulfilled their own philosophical prophesy: They can validate neither the foundation of their science nor the meaning of their methods.
Some mathematics professors are disturbed by this, and-to their credit-continue to seek a solution. Some try to ignore the fact and do their applied work as best they can based on “intuitive” notions-i.e., ideas that are not really understood. Others seem complacent or even glad, feeling that the failure of modern “axiomatic theories” merely underscores that their fellow post-Kantians are right-that we cannot really know anything. In this way, the post-Kantians have tried to turn their mathematical failure into a philosophical victory.
The acceptance of a rational, philosophically validated theory of mathematics would defeat the post-Kantians on two important battlefields. It would rescue mathematics from their grasp; and it would re-establish mathematics as a great example of reason’s power, not its alleged futility.
Defeating philosophical villains, however, is only my secondary purpose in this article. My primary purpose is to elaborate on a positive theory of mathematics that I hope will further the understanding of the subject by everyone from students, to laymen, to advanced professionals.
Mathematics in One Lesson (Part II)
by Ronald Pisaturo
October 1998, Vol 12 No 10
In Part 1 of this article, I showed how the principle of the uniform unit explains multiplication, exponents, and irrational numbers. Now, let us examine two other important kinds of numbers: negative numbers and complex numbers.
Comprehending the principle of the uniform unit is the key to understanding every method of mathematics. Finding a uniform unit is the key to applying mathematics to a specific science. Finding a new kind of uniform unit, or a new way of dealing with uniform units, is the key to devising a new mathematical method.
To understand mathematics, remember this one principle: the uniform unit.
Undermining Reason: The 20th Century’s Assault on the Philosophy of Mathematics, Part 1
by Ronald Pisaturo
October 2000, Vol 14 No 10
In 1998 one philosopher gave the following accurate summary of the end-result of the work on the philosophy of mathematics in the 20th century:
At the [previous] turn of the century, philosophers and mathematicians, reacting to the set-theoretic paradoxes, undertook programmes designed to place mathematics on an unshakable foundation. A century later, philosophers have largely abandoned those dreams.
Advanced mathematics has been used to send spaceships to the Moon and Mars, but it is still not understood why “1 + 1 = 2” is a true statement about reality. And worse: The attempts to show that mathematics represents knowledge of reality have been so unconvincing that there is now widespread doubt that mathematics is true.
This doubt about mathematics has undercut confidence in all human knowledge. Historian of mathematics Morris Kline writes:
The attainment of apparent truths in mathematics and mathematical physics had encouraged the expectation that truths could be acquired in all other fields of knowledge…. The hope and perhaps even the belief that truths can be obtained in politics, ethics, religion, economics, and many other fields may still persist in human minds, but the best support for the hope has been lost. Mathematics offered to the world proof that man can acquire truths and then destroyed the proof.
Modern, post-Kantian intellectuals (of which Kline is one) cite the “failure” of mathematics as further “validation” of skepticism and subjectivism. In truth, however, these post-Kantians are the very cause of the failure they cite. The failure to demonstrate the objectivity of mathematics is, in reality, due to the Kant-inspired philosophic premises at the root of today’s predominant approaches to mathematics.
My previous articles on mathematics in TIA focused on presenting a positive theory of the philosophical foundation of mathematics rather than critiquing other theories. In so doing, those articles showed how mathematics can be rescued from the grasp of the post-Kantians, re-establishing mathematics as a great example of reason’s power, not its alleged futility.
But it is also important to understand the dominant false theories at the philosophical foundation of mathematics. Such knowledge will certainly be useful to aspiring professional mathematicians, who will encounter these ideas in their studies and will be expected to accept them as non-controversial.
But more broadly, an analysis of the collapse of the philosophy of mathematics offers a case study, understandable even to laymen, of the need for a proper theory of concepts-by showing the disasters that result from that theory’s absence.
Undermining Reason: The Assault on the Philosophy of Mathematics, Part 2
by Ronald Pisaturo
November 2001, Vol 15 No 11
Since the late 19th century, several important attempts have been made to defend mathematics–to establish an objective base for the science. One of the most famous of these attempts–still quite popular today, and taught in virtually all first-level abstract-mathematics courses–is the “axiomatic system” published by Giuseppe Peano in 1889.
At the time of its publication, mathematicians were struggling to understand the precise conditions under which their mathematical methods, especially the methods of calculus, would apply. For example, mathematical methods for approximating the value of functions were valid only for certain kinds of functions; therefore, mathematicians needed to know exactly what conditions a function must satisfy in order for the method to work. Sometimes, mathematical theorems “proved” by esteemed mathematicians turned out to be false; the theorems were found to have contained errors in reasoning. The methods of proof in calculus had not been as fully and objectively codified as they had been in algebra and geometry. Peano himself had become well-known for finding counter-examples to alleged theorems that had been accepted as true.
The more that mathematicians struggled with objectifying their methods, the more they saw a need to go back to first principles–all the way back to the basic nature of the simplest numbers: the counting numbers (also known as “natural numbers”): 1, 2, 3, ….
Undermining Reason: The Assault on the Philosophy of Mathematics, Part 3
by Ronald Pisaturo
December 2001, Vol 15 No 12
Peano was not the only thinker of his time who tried to state and validate the basic premises of mathematics. Peano’s contemporary Gottlob Frege (1848-1925), today widely held among mathematicians to be the second greatest logician ever (after Aristotle), pursued the same goal in a series of books published between 1879 and 1903. This commonality with Peano’s goal is apparent from the beginning of the preface to Frege’s 1879 volume:
In apprehending a scientific truth we pass, as a rule, through various degrees of certitude. Perhaps first conjectured on the basis of an insufficient number of particular cases, a general proposition comes to be more and more securely established by being connected to other truths through chains of inferences…. Hence we can inquire, on the one hand, how we have gradually arrived at a given proposition and, on the other, how we can finally provide it with the most secure foundation.
Indeed, Peano and Frege corresponded and saw important commonalities in their approaches as well as their goal. In an 1894 letter to Frege, Peano wrote, “From the notes I sent you … you will see that we are taking the same route in science.” But while Peano sought axioms that were facts of reality and was only drawn eventually into formalism by his epistemological errors, Frege’s theory was severed from reality at the start.
In my articles, I take what I think is a distinctive approach to identifying the axiomatic base of mathematics. I argue that the sole distinctively mathematical axiom is the premise that all of the units being considered are equal–i.e., interchangeable–in the context being considered. Of course, this premise is consistent with the ancient Greek concept of “arithmos” and with Cantor’s idea of “cardinal number” as the result of a “double act of abstraction”; but I identify this premise explicitly as an axiom–indeed, as the axiom–of mathematics. I state this axiom in the form, “1 = 1”, which means not that some abstract object named “1” is equal to itself, but rather that each unit being considered is interchangeable with each other unit. (J.S. Mill also claimed that “1 = 1” is the basic premise of mathematics, but he doubted the truth of the premise.) E.g., each meter is interchangeable–with regard to length–with each other meter.
I claim that, from this one mathematical axiom, and using the principles of inference known from philosophy (which is more basic than mathematics), all of mathematics is developed and can be validated.
The articles are decidedly non-technical. Indeed, undergraduates and laymen, as well as professors of mathematics, have written to tell me that the articles have significantly enhanced their understanding of mathematics.