From the Philosophers

As detailed in my last post on Education vs. Job Training, the real importance of mathematics education is to free the mind from the illusory way of knowing derived from the senses and empower the conclusion-forming process so that truth as such is at last discernible.  So that readers may understand that this is not a new idea but actually an old one, I will introduce my entourage of philosophical luminaries and present their words as evidence that this is how mathematics was intended to be learned.  Working backwards through history, we shall see that the clarity and conciseness of expression is the greatest when we consider the earliest of the sages, thus demonstrating that the original view of mathematics is the one I am advancing here.

First let us consider the wonderful summary of Rudolf Steiner (1861-1925), as he goes into great detail on the necessity of mathematical studies as a prerequisite for entrance into the Academy.  This selection is from a 1904 lecture by Steiner.

Rudolf Steiner (1861 - 1925)

"It is well known that the inscription over the door of Plato's school was intended to exclude anybody who was unacquainted with the science of Mathematics, from participating in the teachings of the Master. Whatever we may think of the historical truth of this tradition, it is based upon the correct understanding of the place that Plato assigned to mathematics within the domain of human knowledge. Plato intended to awaken the perceptions of his disciples by training them to move in the realm of purely spiritual being according to his “Doctrine of Ideas.” His point of view was that Man can know nothing of the “True World” so long as his thought is permeated by what his senses transmit. He demanded that thought should be emancipated from sensation. Man moves in the World of Ideas when he thinks, only after he has purged his thought of all that sensuous perception can present.

"It was precisely such a mind emancipated from sense-perception and yet spiritually full, which Plato demanded from those who would understand his Doctrine of Ideas. In demanding this, however, he demanded no more than was always required of their disciples, by those who aspired to make them true initiates of the Higher Knowledge. Until Man experiences within himself to its full extent what Plato here implies, he cannot have any conception of what true Wisdom is. 

"The Gnostics desired something similar. They said, “Gnosis is Mathesis.” They did not mean by this that the essence of the world can be based on mathematical ideas, but only that the first stages in the spiritual education of Man are constituted by what is supersensible in mathematical thought. When a man reaches the stage of being able to think of other properties of the world independently of sense-perception in the same way as he is able to think mathematically of geometrical forms and arithmetical relations of numbers, then he is fairly on the path to spiritual knowledge. They did not strive for Mathesis as such, but rather for supersensible knowledge after the pattern of Mathesis. They regarded Mathesis as a model or a prototype, because the geometrical proportions of the World are the most elementary and simple, and such as Man can most easily understand.  In this way did Plato and the Gnostics conceive mathematical science as an educational means."

The words of the English Platonist Thomas Taylor (1758-1835) can give us a very erudite detailing of the difference between mathematics as job training and mathematics as education, in the introduction to his Theoretic Arithmetic of the Pythagoreans.  Here Taylor uses the word "arithmetic" for what we today would call Number Theory.

Thomas Taylor (1758 - 1835)

"The mathematical disciplines have been rather studied with a view to the wants and conveniences of the merely animal life, than to the good of intellect in which our very being and felicity consist.  This observation particularly applies to Theoretic Arithmetic, the study of which has been almost totally neglected: for it has been superseded by practical arithmetic, which though eminently subservient to vulgar utility, and indispensably necessary in the shop and counting house, yet is by no means calculated to purify, invigorate, and enlighten the mind, to elevate it from a sensible to an intellectual life, and thus promote the most real and exalted good of man.

"Mathematics' first and most essential use is that of enabling its votary, like a bridge, to pass over the obscurity of a material nature, as over some dark sea to the luminous regions of perfect reality; or as Plato elegantly expresses it, "conducting them as from some benighted day, to the true ascent to incorporeal being, which is genuine philosophy itself." 

"True mathematics is for the most part sordidly neglected, because it neither promotes the increase of a commerce which is already so extended, nor contributes anything to the further gratification of sensual appetite, or the unbounded accumulation of wealth.  If the mathematical sciences, and particularly arithmetic and geometry, had been studied in this partial and ignoble manner by the sagacious Greeks, they would never have produced a Euclid, an Apollonius, or an Archimedes, men who carried geometry to the acme of scientific perfection, and whose works, like the remains of Grecian art, are the models by which the unhallowed genius of modern times has been formed, to whatever mathematical excellence it may possess."

Taylor then invokes Plato and his philosophy of ideas, and how mathematics leads us into this higher world of being.

Plato

"Plato calls the knowledge of the mathematical disciplines the path of erudition, because it has the same ratio to the science of wholes, and the first philosophy, or metaphysics, which erudition has to virtue.  For the latter disposes the soul for a perfect life by the possession of unperverted manners; but the former prepares the reasoning power and the eye of the soul to an elevation from the obscurity of objects of sense.  Hence Socrates in the Republic rightly says that "the eye of the soul is blinded and buried by other studies, is alone naturally adapted to be resuscitated and excited by the mathematical disciplines."  And again, that "it is led by theses to the vision of true being and from images to realities, and is transferred from obscurity to intellectual light, and in short is extended from the caverns of a sensible life and the bonds of matter, to an incorporeal and impartible essence."  For the beauty and order of the mathematical reasonings, and the stability of the theory in these sciences, conjoin us with and perfectly establish us in intelligibles, which perpetually remain the same, are always resplendent with divine beauty, and preserve an immutable order with reference to each other.

"He, therefore, who is naturally a philosopher, is excited indeed from himself, and surveys with astonishment real being.  Hence, says Plotinus, he must be disciplined in the mathematical sciences, in order that he may be accustomed to an incorporeal nature, and led to the contemplation of the principles of all things.  From these things, therefore, it is evident that the mathematics are of the greatest utility to philosophy."

Again Taylor returns to the theme of Education vs. Job Training that we have been circling around in these posts.

"But we ought to judge of its utility, not to the conveniences and necessities of human life.  For thus also we must acknowledge that contemplative virtue itself is useless.  For Socrates says that through intellectual energy the philosophers are separated from all habitude to human life, and from an attention to its necessities and wants, and that they extend the reasoning power of the soul without impediment to the contemplation of real beings.  The mathematical science, therefore, must be considered as desirable for its own sake, and for the contemplation it affords, and not on account of the utility of administers to human concerns.  If however, it be requisite to refer its utility to something else, it must be referred to intellectual knowledge.  For it leads us to this, and prepares the eye of the soul for the knowledge of incorporeal wholes, purifying it, and removing the impediments arising from sensible objects.

"As therefore, we do not say that the whole of cathartic or purifying virtue is useful, or the contrary, looking to the utility of sensible life, but regarding the advantage of the contemplative life; thus also it is fit to refer the end of mathematical science to intellect, and the whole of wisdom.  The energy about it deserves our most serious attention, both on its own account, and on account of an intellectual life.  Those who despise the knowledge of the mathematics, have not tasted of the pleasures they contain.  The mathematical science, therefore is not to be despised, because its theoretical part does not contribute to human utility; but on the contrary we should admire its immateriality, and the good which it contains in itself alone."

We shall have occasion to return to the philosophers in future discussions, for now this should be enough to see the pattern developing.

Job Training vs. Education

Why do we need to have mathematics in the college curriculum?  Is it because students need to acquire quantitative skills that they might use on the job some day?  It would certainly seem that way judging from the way our textbooks are written.  The prevailing opinion seems to be that since it is possible to create a scenario where a solution to a problem arises from solving an algebraic equation, that we should teach everyone these "skills", just in case the student is faced with this problem on the job some day.  I would like to make a case for why this kind of thinking takes all the value out of mathematics, and actually detracts from the true purpose of mathematics education.

Take for example the problem of finding the break-even point in a cost-revenue analysis.  The problem is to find the production level for which cost equals revenue.  This is a problem that may come up in an actual business situation.  But what kind of person will need to answer this question?  Probably not a person who is only required to know a small amount of algebra to earn their required minimum math credit.  So why would we teach this type of problem to everyone, regardless of whether their degree is in business?

If someone needed to know how to do this, it should be taught in a business class.  Actually any smart business would have already solved this problem and established a formula so that their numbers could simply be plugged-in to get the break-even point, and no one would actually ever write down an algebraic equation and solve for x as part of their daily routine.  There's really no need to teach everyone how to solve the problem if all we care about is having a technique to get the answer.  But it seems that there's still a good reason to teach someone how this problem was solved in the first place.

Learning the problem of finding the break-even point is actually a fantastic example of how our minds work, and can demonstrate the process of problem solving.  But it is not taught as such, the only emphasis being "you may be faced with this problem some day, and here's what you do."  In other words, we have completely removed the education experience and replaced it with job training.  Yet it is depressing to students to think that they have to struggle learning all of the language, notation, and mechanics of algebra just so that they may need it for the job.  No wonder we as teachers are seeing such depleted morale in our classrooms.

Wouldn't it be better to teach students how to think, not what to think?  If you were truly educated, and knew how to think, you wouldn't need to be shown what to do to solve this problem, because you could think for yourself and apply problem solving skills.  You would be a human being with a mind, not a robot who does exactly as told.  This is the essence of the problem.  Job training tells people what to think, education teaches people how to think.  Remember the old "give a man a fish..."?  As an educator I would prefer to teach people how to think, and thus feed them for a lifetime. 

Administrators who need to make "fiscally responsible" decisions realize that they too have a break-even point that needs to be met.  Unless a certain "production level" of students is met, the business is operating at a loss.  "Come to college, get a great job, and now you're rolling with cash" is their hypnotic chant.  What better selling-point than this could be advertised to lure in prospective customers?  But the business of college cannot represent itself this way as an institution, so they hide their true money-making agenda under such "mission and values statements" that claim education is the top priority.  I would like this to be true, but as a teacher I am required to teach specific things that are not the best for imparting a top-quality education experience.  It seems that colleges and their government financiers do this because they don't really want well-educated citizens who can think independently, probing for the truth, just workers who are good at taking orders.

"The mind is not a vessel to be filled, it is a fire to be kindled."    
-- Plutarch, Greek Platonist, 1st century C.E.

It surprises me however that, while some colleges do have better ideals and take leading initiatives to increase student success and graduation rates, they don't realize that it is their view of the role of mathematics in the curriculum that creates a reversal in the students' motivation for success.  "Why do they make me take these math classes?  I just want a good job, I'll never use this stuff", which is what anyone would think who thought they were getting a worm for a meal and now has a sharp hook in their mouth.

All criticism aside, I think the problem is simply that no one but a true mathematician can understand why we need mathematics, and usually a true mathematician does not become a college administrator.  I have decided to use my understanding of the original intention of mathematics to campaign for mathematics as education and not as job training.

Mathematics, rightly understood, is completely useless.  "If it ain't got no use, why do we need to learn it?"  Here I'm merely resounding an echo that causes pain in my ears.  Actually I'm playing upon the meaning of the word "use", because in the sense it appears here, it refers to the vulgar use of mathematics, which is all most people can conceive.  The vulgar use of mathematics refers to building houses and other such things. This is how you think when your mind is clouded by a false belief that the material world is all there is.  We must understand that mathematics was not invented by the Greeks for building houses, but for building temples.  This is actually a metaphor that shall now be expounded. 

The true enlightened use of mathematics is to liberate the mind from the illusory way of thinking derived by sense experience.  It is meant to purify the mind from accepting appearance as reality, and to lead it into a way of knowing the truth as such.  Unless one builds knowledge systems after the pattern of mathematical knowing, it is inherent that errors will manifest themselves.  Only one who thinks like a mathematican can become aware of the subtleties of reasoning, and can establish truth on a rigorous foundation.

Even if we don't want to become philosophers, although really we should, since if we are not lovers of wisdom then we are not experiencing the fullness of life, there are still good reasons for why true mathematics is a necessary component for every person who claims to be educated.  Mathematics teaches us how to think, not what to think.  It teaches us reason and logic, problem-solving, and how to discern error from truth.  If you can't think for yourself, you aren't free.  If you always have to have someone else do the thinking for you, you are still a child mentally.  Many of the problems we are faced with in today's world are due to masses of so-called educated people not being able to think for themselves, and relying on elected individuals to do the thinking for them.  Much of the problem could be reversed through proper mathematics education, which actually makes you want to think for yourself and not let others do it for you, and soon you learn to love doing this.

Mathematics is full of great use, but this perspective has been drowned out through gradual historical neglect.  The use of mathematics for freeing the mind and knowing the truth was established by the Greek philosophical elite, such as Pythagoras, Socrates, Plato, and Aristotle.  They knew the difference between the vulgar use of mathematics versus the enlightened use I am promoting here.

Why did Plato have the words "Let no one ignorant of geometry enter" chiseled over the portal to the Academy?  Were they busying themselves with calculating land areas and how much wood was needed to build a house?  Why did you need to know geometry to hang with the philosophical elite of Athens?  My next post will highlight the specific words of the philosophers and show that this was their understanding of mathematics education.  I leave you with this sample as an indication of where we're headed:

"My noble friend geometry will draw the soul towards truth, and create the spirit of philosophy."
-- Plato, The Republic