More Vague Notes on Basic Algebra as Expression Arithmetic

In my last post I said that it was common knowledge among mathematicians that basic algebra is best described as expression arithmetic. I'm continuing my monologue on these ideas here.

Addition, as an operation of elementary arithmetic, is a procedure that takes a pair of decimals and gives a decimal that represents the sum of the numbers denoted by the taken decimals. It is often frowned upon within some mathematical circles to describe addition in this way, as it requires an overly pedantic distinction between decimals and the numbers that they represent. I can not agree with this perception, nor can I allow it to be propagated, it is not pedantic. In fact, it cuts to the heart of the mathematical method: represent a problem in a form that you can understand and easily manipulate.

The theory of cardinals is rich and deep, it requires a lot of work to prove that the sum of a pair of cardinals is another cardinal, yet in the case of finite cardinals, we need only know that the arithmetic we've developed is an adequate representation of the theoretical backings. If we are curious, and we should be, we will eventually push past the function of arithmetic towards the form of cardinals so that we might see how and why arithmetic is done as it is.

Children, on the other hand, do not need to follow the path of logic through modern mathematical topics. If we can give them tools for thinking about mathematics efficiently then we can leave it to their individual curiosity to look for deeper meaning and form. Not every person will have an impulse to understand the way things work, though they probably could have such an impulse if they were properly educated. Some people will naturally be compelled to follow the facts past what they've been told towards the truth that they are able to demonstrate themselves.

It benefits both types of people to learn efficient methods of dealing with mathematics that might otherwise have deep underpinnings. Both will learn to use math, and the one who is interested in going deeper will have already developed an effective intuition as to what principles and problems are of central importance in the use of mathematics as a tool for thought. Finally, if we are lucky, the one who searches for truth will discover that there is more to mathematics than its utility as a tool for thought, they will be compelled, we hope, to dream of ever more beautiful mathematics that eliminates the illusion that utility and ornament are disjoint.


Vague Notes on Elementary Algebra as Expression Arithmetic

Though it is common knowledge within the mathematical community that elementary algebra is best described as the arithmetic of basic algebraic expressions, the consequences of this perspective have yet to penetrate into the elementary level. Not every description of a subject impacts its practical use, though it is often better to have multiple descriptions of a familiar object, each of which reveals some feature that might otherwise be obscured in alternate descriptions.

That elementary algebra is an arithmetic of expressions is of great practical importance. It not only gives a sense of what elementary algebra is, but also how one should go about developing its skills. A student of elementary algebra is often mystified by the use of variables, something which continues to be misunderstood by professional and nonprofessional mathematicians alike. By viewing expressions as nothing more than an extension of the numerals used in elementary school, variables become just another thing to play with, not some sort of mystical quasi number or some other such equally vague notion of "unknown".

The fact which is of greatest importance, and most often ignored in our modern presentations of algebra, is that just like the distinction between numbers and numerals, expressions are just special assemblies of marks that we may interpret and operate on without tending to their meaning in a specific context. We do not need to know what number is in order to operate with numerals, and we do not need to know what sentential functions are in order to operate with expressions. In fact, our knowledge of number is informed by our use of numerals, and the same is true of sentential functions.

This wasn’t always the case. People thought they were dealing directly with number when what they were really working with were numerals, some more efficient in their use than others. In general, we tend to work with physical representations of mathematical concepts because, as humans, we have a certain intuition for naive spatial relations and motions. The extent to which there are physical representations of mathematical concepts that faithfully translate the features of the original mathematical concepts often separates good notation from bad notation.

Perhaps the most important reason to reduce algebra to an arithmetic, is that an arithmetic is by its very essence a physical representation of a mathematical concept. Often an arithmetic is a set of procedures performed on a collection of marks. From this perspective we might as well call an arithmetic a set of instructions or algorithms for operating on assemblies of marks.

Furthermore, it is now completely satisfactory to operate as if logic is itself a special algebra. This perspective has been used with great benefit both practically and theoretically. By interpreting logic as algebra, and algebra as arithmetic, we quickly see how it might be possible to place ever more abstract concepts into the hands of unknowing children by properly translating things into an appropriate arithmetic. If a child can learn elementary arithmetic, then they ultimately have all the mathematical tools they will need for the rest of their mathematical lives, what they lack is a certain set of problem solving skills that are not strictly mathematical (though it is often easier to develop problem solving skills in general by tending to mathematical problems whose content are much more precise than the vague problems we are confronted with in everyday life).


A Change in Tone

I am not only changing the tone of this blog from formal to informal, but I am also going to widen the scope of topics covered here. In my natural habitat I’m friendly, fun, and informal. So, rather than working against myself, I intend on embracing those characteristics of my personality that shine throughout my everyday life.

Math, and science, are, for me, a part of my deep, and seemingly instinctual, love for knowledge. I actually enjoy knowing things, sharing what I know, and learning new things. Rather than sitting in front of a television to watch the latest sitcom or picking up a game controller, I’m likely to spend my free time reading and taking notes on math and physics because it makes me happy to do so. What gives me the greatest joy is understanding something that was once vague and imprecise in terms that are not only simple but absolutely clear. Once I understand something well, then I am often able to explain it to people in a way that is both fun and functional.

Unlike most people who enjoy knowing things, I do not feel that much satisfaction in knowing things without knowing how to share them. That means that when I take notes, or try to figure things out on my own, I’m really trying to figure things out in a way that is blindingly obvious and is connected to relevant features of day to day life. Things that dangle from a long chain of abstraction have got to be connected to something that I can make sense of, something that I can think of in terms of a well defined problem, or something which helps me to better organize my knowledge of the world, otherwise I loose sight of it among the multitude of meaningless abstractions one can build up with just a bit of mathematical imagination.

What does this mean for the blog? I’m not exactly sure yet, but I do know that rather than being a collection of well constructed facts, or a presentation of some new connection between facts, I will be posting more frequently about things that I’m collecting, not just those things which I’ve already organized and subsequently revised.

I honestly think that mathematics is a practical tool for living and working in this world. Most people think this means that I want to use math in order to engineer a better sprocket or create the next internet. This is common because not too many people are familiar with the methods of mathematics, they only know of its products. When mathematicians open up a new problem, they go through a process of identifying similarities with previous mathematical problems as well as any relevant figures or illustrations that draw up the most vivid and inspiring associations. That process of making sense out of the impossible is what really draws me to mathematics and continues to bring me back to the problems of this world.

Most, if not all, of the problems in this world can be remedied with clarity and exactness. Rather than standing in a vague haze of words that connect only to half-facts, we must move ourselves closer towards the vague outlines that are presented to us until we see something that is distinct or clear in form. Without clarity, we tend to argue over nonissues, nonfacts, and nonproblems. The path from vagueness to clarity is rugged, and, like any good mountaineer will tell you, one does not simply walk up to the top of Mount Everest on a whim. We traverse rugged terrain by acclimating ourselves to the environment through successively more difficult outings. The problems of mathematics are something of a rose garden in this way. They are often well organized, well traversed, and the beauty is almost self evident no matter how far from a solution you are (that is if you have a good mathematical gardner keeping watch).

One day I hope to develop the skill needed to write with that proper combination of formal and informal tone. For now, I will allow myself to be satisfied with an informal tone that, I hope, will occasionally dip into something more formal when I’m inspired or when the material demands it.


A Place to Call Home

Dear Reader,

I want to bring warmth to human inquiry. The cold and desolate land of useless facts and figures is of little interest to all but the most lifeless of minds. Those who find the greatest joy in honoring their love of knowledge have successfully obtained that long forgotten happiness which still lurks in the most innocent corner of our minds. The strength of curiosity in a child is a testament to the intrinsic value of a distinctly joyous pursuit of knowledge. Once a child has acquired some fact or skill they are apt to repeatedly display it and to rejoice in their mastery of what once seemed a mystical art from the unknown.

For my part, I have lost what was once the fun of learning and discovering, and this disgusts me. I have spent my life in pursuit of knowledge and wisdom only to find my love questioned, ridiculed, and marginalized. I have found myself wondering where the joy has gone: when did the mastery of a new intellectual skill loose its luster?

As I climbed higher and higher within the world of academics I became aware of an ever growing sense of combative rather than cooperative competition. I saw others acting primarily out of an overpowering fear of failure rather than an overpowering love of knowledge. That success was measured by academic excellence was an unquestioned axiom. School seemed a place for turning intellect into income, and my mind began to tear itself apart searching for its place among such tumult.

I am not a politician and as such my innocence was crushed by the weight of worry and absolute confusion. How had I educated myself out of my element? Was I to blame for arriving at a place where my interests could not flourish? The only answer which presented itself was yes. I took all of the actions that developed my mind into what it is, and it was this that led to my personal demise. I was responsible for undoing my own intellectual interests.

There was a time when my concern for the future of my happiness became so great that I had tricked myself into believing that my mathematical interests were the principle source of my personal sorrow. It was a dark time, and I survived it only with the help of family and friends. But how, among all that I had done, was I to explain to myself the state I was in? Had not my skills in mathematics and physics brought me to this sorrow?

For a while I denied myself the content of my curiosity and interest so that I might find some much needed comfort in conforming to the expectations of my position within society. Had I been wiser at the time I may have foreseen the cost of such an impossible temperament. I had taken the war going on outside of me and placed it squarely within my own mind.

At every turn of a new idea my desire for a deeper understanding grew with a fierce intensity which pained me to ignore. I had thought that the wisest course of action was that given to me by others: work against your interests until you might some day gain the power to set them free. I was a fool for even imagining that I might follow such a path, much less emerge from my journey unscathed. Though I seem to have avoided any mortal wounds, to this day I have scars which never cease to remind me of the dangers of denying the natural impulses of my intellectual interests.

I escaped what may have been a horrible end by leaving behind my dreams of academic excellence. I now know that there is nothing unique about my position and path, and this is why I find the entire sequence of events hideous. How, among the many advances that have been made in our modern society, can someone like myself find such sorrow in that which once brought such joy?

As any reasonable person would have done, I sought possible origins of my almost unnatural unhappiness. “Perhaps I suffer from severe depression?” I thought. After having a few anxiety attacks I pursued this line of inquiry with vigor. I discovered that even after managing my anxiety the weight of my sorrow had not been lifted. Worse, it seemed to have become contagious. Those around me who had entered the academic world with interest and intrigue appeared to be suffering as I was. The joy had been removed not only from their minds but also their hearts: you must hide your moments of excitement lest they interfere with your own self deception.

I left all that I knew to save myself from a never ending pit of sadness. I have since discovered fragments of that which I thought had been permanently lost, but even now I have not found my place in this world. Rather than wait for a place for my mind to call home, I have decided to build my own place in this world. I may not be nearly as certain, as I once was, in where I might find my sense of personal success, but I certainly can not let myself live with the fear of failure that is sure to stifle any sense of joy I might get from this life.


John Meuser


Some Reflections on Kleene's Introduction to Metamathematics and Bourbaki's Theory of Sets

Recently I have been reading Kleene's Introduction to Metamathematics and have found his descriptions of the various foundations for mathematics exciting. I am excited not only to find such well written descriptions of the methods of metamathematics, but also to find another mathematician whose beliefs appear to be quite close to my own, at least in his description of metamathematics. I do not claim to know to what extent his beliefs about non-mathematical things are in line with my own as I have found through experience that a mathematicians beliefs about mathematics, though they should logically have a strong influence on their beliefs about non mathematical things, tend to have a sporadic impact if any on their non-mathematical beliefs.

His text has been inspiring both because of its clear descriptions of formal mathematics in a way which is much more prescient than the others that I have read, and noticeably accessible in its accommodations of a wider scope of investigation into intuitionistic logic, and because of its support of my belief that there must be greater effort put into describing the mechanics of mathematics, that is the details of the actions which are themselves mathematical. Specifically, the recognition that to the formalist the objects of mathematics are certain texts which represent a formal description of some area of mathematics without appealing to the meaning that is usually attached to the objects described by the formalism. As much as mathematics is meant to discover things which are timelessly relevant it would seem the formalists have made a key observation which will persist, in some form, for at least the next century if not the next millennium.

Unsurprisingly, while I've been reading Kleene's IM I have been comparing its observations to Bourbaki's Theory of Sets. Much of the criticism of the work of Bourbaki is focused on its Hilbertian foundations. There are also those who dislike that Bourbaki only mildly mentions the work of Gödel, Kleene, Bernays, Cohen, Church, Schoenfield, Tarski, Brower, Gentzen, Skolem, and all the other mathematicians, logicians, and mathematical logicians who have contributed to our modern understanding of the foundations of mathematics.

The claim that Bourbaki's Theory of Sets is Hilbertain is strict in that it uses a form of Hilbert's Epsilon Calculus as its logical foundation. Instead of using Hilbert's \(\epsilon\) notation, Bourbaki uses a distinct \(\tau\) and \(\square\) notation with linkages. Informally, if \(R(x)\) is a relation and if there is an object which satisfies the relation \(R(x)\) when substituted for \(x\) then \(\epsilon x R(x)\) represents such an object, otherwise \(\epsilon xR(x)\) is a thing about which nothing meaningful can be said. Again, this vague description is not meant to be anything other than an informal description. I do not yet know enough about the development of Hilbert's epsilon calculus to make a formal comparison between its use in his program, and its use in Bourbaki's theory of sets.

Though I am not deeply familiar with Hilbert's epsilon calculus in general, I am familiar with the logic of Bourbaki, and find it interesting that the logical symbols \(\tau\) and \(\square\) are used, together with an informal notion of substitution, to define the existential and universal quantifiers. Additionally, the modern role of the axiom of choice is reduced entirely to certain statements involving the \(\tau\) notation. This is yet another point of contention with most modern mathematical logicians, as some believe this method "hides" the use of the axiom of choice in the formal language rather than as an additional axiom on a footing equal to the pairing axiom or power set axiom.

As far as Bourbaki is concerned, the purpose of their description of formal mathematics is to avoid circumlocutions leading up to their development of their theory of sets. Bourbaki does not set out to show how their theory of sets sits among the differing foundations of mathematics, they simply wish to get as quickly as possible to a description of sets so that they can proceed with their treatment of the rest of modern mathematics. Said another way, it is not the purpose of Bourbaki to formalize mathematics, but rather to present it as being describable using the basic notions of set theory. The belief that mathematics may be almost entirely based on set theory is often given the hyperbolic name "Cantor's Paradise" so as to give credit to Cantor's independent discovery and creation of the precursors to modern set theory.

It is admitted by Bourbaki that their informal goal is to use set theory as a way of classifying and organizing modern mathematics. Specifically, the final chapter of Theory of Sets is entitled Structures and the second section is entitle Species of Structures. Though the definition of structure is given formally, it includes the formal language of set theory developed earlier in the book. The basic idea is that a structure is an element of some set which is built from the fundamental constructions permissible in Bourbaki's set theory. For example, within the classical theory of sets a topology \(\mathcal{T}\) on a set \(X\) is a collection of subsets of \(X\) which satisfy two relations:

  1. if \(\mathcal{U}\) is a subset of \(\mathcal{T}\) then \(\bigcup \mathcal{U} \in \mathcal{T}\); and
  2. if \(\mathcal{U}\) is a finite subset of \(\mathcal{T}\) then \(\bigcap \mathcal{U} \in \mathcal{T}\).
Thus, a topology on a set \(X\) belongs to \(P(P(X))\), the power set of the power set of \(X\). Furthermore, the collection of all topologies on a set \(X\) is a subset of \(P(P(X))\) whose members satisfy these two topological relations. Consequently, the set of all topologies on a set \(X\) is a unique member of \(P(P(P(X)))\), so that what it means to be a topology on \(X\) is governed in part by a specific set of axioms and membership to a set constructed from \(X\) by iterating the power set operation.

A somewhat simpler example is the structure of relations on a set, which are completely described by being a member of the cartesian product of the set under consideration. That is, a relation on a set \(X\) is a member of \(X \times X\) and the collection of all relations on a set \(X\) is \(P(X \times X)\). So that \(R\) is a relation on a set \(X\) if and only if \(R \in P(X \times X)\). Here it is clearer that what it means to be a relation, or what the structure of a relation is, on a set \(X\) is characterized by the construction of the power set of the cartesian product of \(X\) with itself.

So the species of structures which are examined in mathematics are, as is assumed by Bourbaki, of this form. Each species of structure has some set or collection of sets as its base, like \(X\) in the previous two examples, and may be specified by writing down to which set constructed from \(X\) the structure belongs as well as the basic axioms that the structure must satisfy.

Much of the detail has been removed from this informal description, and modern mathematicians have continued to develop these ideas, though it still seems that the notion of structure is meaningless outside of a formal description of mathematics. The modern notion of structure has been taken back from the theory of sets by the model theorists (and universal algebraists) so as to not only permit varying structures of set theory, but also provide a more comprehensive framework for structures describable using a formal language which might not contain a theory of sets. In this context, specifically the model theoretic, a structure is more commonly referred to as an interpretation.

This push towards the logical foundations of mathematics brings me back to my interest in Kleene's IM. Though Shoenfield's Mathematical Logic is considered the modern introduction to mathematical logic, Kleene's IM, as well as his own Mathematical Logic which may be bought as a Dover book, is a much slower and digestible than Shoenfield's terse and dense book. Both are beautiful and for those interested in modern mathematical logic the two texts, Kleene's IM and Shoenfield's ML, are required reading. One important component of Kleene's IM which is sadly missing from Shoenfield's ML is Intuitionistic Logic. Kleene gives a description of the three most well known approaches to the problem of finding a foundation for mathematics: the logicists of which Russell and Whitehead are the principle members, the formalists of which Hilbert is most well known, and the Intuitionists of which Brouwer is the most well known.

Though each school of thought is based on a differing set of beliefs each shares some characteristics with the others. In particular, the use of formal languages may be identified across each school. Though, it is interesting to note that Gödel's monumental paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems is rooted in his recognition that the logical mechanisms employed by Russell where not formally described, and that once they were certain questions of consistency and completeness naturally arose. Russell's description of mathematical logic was of a decidedly philosophic nature, occasionally relying on informal descriptions rather than formal syntactic rules, as he believed that his foundation of mathematics was a description not of a philosophic ideal, from Plato, but rather a tool for describing that section of philosophic logic which results in our modern study of mathematics. His thesis, that mathematics is reducible to logic, is often misunderstood as referring to the modern use of formal texts to describe mathematics. Rather, Russell would clearly identify formal texts as being part of mathematics, and the type of thing which must ultimately be justified from a philosophically defendable position, one which necessarily requires the acceptance of some metaphysic and logic which itself need not be a formal mathematics.

The distinction between the formal and informal components of mathematics is expertly revealed in Kleene's IM. It is the principle reason his exposition is so familiar to modern mathematical logicians. The study of metamathematics is the formal study of formal mathematics, which, apparently, requires the use of informal mathematics. The use of certain fundamentally informal mathematical notions forms the center piece of Brouwer's intuitionism, wherein he believes that the only mathematical arguments which are admissible are those which are supported by a sort of informal acquaintance with certain features of the natural numbers. One of these informal notions is that one can not realize a completed set of natural numbers. Thus a proof of a statement "for all natural numbers \(n\) it is that \(P(n)\)" where \(P(n)\) is some statement about natural numbers, must proceed without the assumption that there is a collection of all those natural numbers \(n\) for which \(P(n)\) is true and a collection of all those natural numbers for which \(P(n)\) is false, otherwise these joint collections would entail the realization of the whole collection of natural numbers. This excludes the use of the classical proof by contradiction which presupposes the existence of such sets. That is, such a proof requires you to accept that \(P(n)\) is "already known to be true or false" of each natural number, something which Brouwer rejects. Obviously these descriptions are informal, hence the value of Kleene's IM.

Kleene's metamathematics gives a framework for dealing not only with classical logics, but also those of the intuitionistic variety. This process is exactly analogous to that employed in the discovery and description of non-Euclidean geometries.

In the future I hope to provide clear and exact descriptions of these informal reflections. The tools for such an exposition appear to be in Kleene's IM and, if not, will likely be found in Shoenfield's ML.