Several religious universities, such as Brigham Young University, ask all instructors in every area to try to make their courses spiritually uplifting. This is something included in student ratings.
When asked to by a department, how can one make a calculus or linear-algebra class spiritually uplifting?
It seems harder to do for math than for history or even economics or physics (I feel that physics, taught right, inspires students with awe for the universe).
For the Christian perspective, a general approach which is useful (these comments apply to any math course which is not merely "plug-and-chug"):
One can explore the spiritual motivations of a variety of mathematicians. Leonhard Euler was the son of a Protestant pastor and was presumably to go into the ministry when Johann Bernoulli intervened. Euler went on to write prolifically about math, science and Christianity (see Letters to a German Princess as an example.) Euler was an outspoken Christian. Newton was a deist and that religious view motivated some of his pursuit to understand the universe. If one wants nonChristian examples, there is Ramanujan (apparently a devout Hindu) and Omar Khayyam (Muslim)….
Many mathematicians were motivated by a view of mathematics as transcendent, touching on the language of God, giving insight into a divine Creation. I think it is fairly natural to touch on this motivation, especially in a religious setting.
It is customary in our rational age to downplay the religious motivations of people like Euler, Newton, Cantor, Ramanujan or Khayyam. Some translations of "Letters to a German Princess" apparently leave out Euler's Christian views; Hardy downplayed Ramanujan's Hindu beliefs and tried to view Ramanujan as agnostic, but I think the truth was otherwise.
For calculus, talk a lot about infinity. This is a topic which is tremendously theologically interesting- from Zeno's paradox (potential infinity) to Rabbi Hasdai Crescas (actual infinity) to Bolzano (a priest by profession, who proved in IVT and saw the necessity for completeness of the real numbers).
The fact that there is more than one infinity is something which divinity students seem to love and get very excited by- and it ties right in with completeness of the reals. So, in a word, if you make calculus rigourous and give a proper discussion of how to construct the real numbers, completeness of the reals, the philosophy behind epsilon-delta (a line as a collection of overlapping neighbourhoods rather than as a collection of points), the definition of a limit, and suchlike, stressing the philosophical insight behind this worldview, I think that the students will find it quite spiritually uplifting!
For linear algebra, I think talking about Hermann Grassmann and his ideas might be a direction to think about, but I haven't thought about it seriously.
Mentioning other mathematician's motivations (as Ken W. Smith did in his answer) might be indeed a good starting point. I'd like to highlight one particular case where beliefs and mathematics come together.
These are the first lines of the introduction of M. Aigner, G. Ziegler: "Proofs from THE BOOK", Springer, 1998 (3rd Ed. - the current one is the 4th, though).
Paus Erdős liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. Erdős also said that you need not believe in God, but, as a mathematician, you should believe in The Book.
I think that this anecdote helps a lot put the concept of beauty and elegance in mathematics into a larger context which touches spiritual topics: For religious students, you have one possible answer to the question "Why is the proof done in this way?" For non-religious students, this might offer an oppurtunity to understand spiritual beliefs better.
On a largely unrelated note: Ziegler and Aigner don't claim to have written THE BOOK - it's just a tribute.
I've never made spiritual uplift a goal of my teaching, but if asked, I would do it using history.
Discuss Newton's General Scholium to the Principia, his greatest application of calculus: "This most beautiful System of the Sun, Planets, and Comets, could only proceed from the counsel and dominion of an intelligent and powerful being."
Discuss Berkeley's Analyst, with its criticism of the unrigorous calculus and frequent comparisons to religion.
Discuss magic squares in a linear algebra class, perhaps reformulating the conditions in terms of matrices.
This has the advantage of connecting spirituality to particular topics in the math classes.
In addition to the above, you might mention that mathematics is the science of material-independent phenomena. On the simple end, three apples, cars, or orang-utans are all "three", but more fundamentally, the same equations describe for example, turbulence, in water, air, or solar plasma.
More: Mathematics demonstrates emergent properties, where simple rules can give rise to incredibly complex structures. Patterns such as the spirals and Fibonacci arrangements of a flower derive purely from the flowers' motions and patterns of growth -- the flower doesn't arrange them, but its buds grow according to simple rules, which produce the patterns. More purely mathematical "emergent phenomena" include fractals such as the Mandelbrot set.
Coming extremely late to this question. I certainly agree with historical approaches of various kinds as being appropriate in this context (presumably also in a "non-spiritually-uplifting context" if that were requested as well) and infinity is always fun.
However, I notice that two issues of application have not been addressed by previous answers. References abound on the internet so I will only put in a couple likely-to-die links.
There is a lot of discussion of the "Unreasonable Effectiveness of Mathematics" article of Wigner's. That article itself has been endlessly discussed, including whether the effectiveness is reasonable, whether there is a reasonable ineffectiveness, and so forth, but at any rate this shows up a lot in discussions of this type, e.g. Chapter 8 of "Mathematics Through the Eyes of Faith", and I think some discussion of it can be "uplifting" to some. Obviously calculus has a lot to do with some of these modeling issues in the physical sciences.
Secondly (and this could cut even into "morally uplifting", it's quite broad) there is the issue of which applications to pursue, and how. Particularly in linear algebra, which underlies so much of modern data analysis/science/AI/you name it, there are real questions of how much to just let PCA give you answers and how much you are asking the right questions. "Weapons of Math Destruction" is a fairly strident take on this, and one might think it's not "uplifting" to dwell on such things - but considering that one has the opportunity to use mathematics to discover places where "the algorithm" has helped perpetuate bias, and combat that, I think this would fit the bill as well. In fact, many humanitarian NGOs now are using tools like linear algebra and OR to help inform their resource allocation to respond even better to crises in the future.
Math, when examined/shown properly, is a beautiful thing, worthy of being called a work of art or natural wonder depending on your stance on things. So it definitely can be spiritually uplifting if taught well.
As to how to teach it like that (i.e. effectively, engagingly and enchantingly), well, if I knew that I'd be shouting it from the rooftops, not here. But speaking as someone who unfortunately was never gifted or particularly proficient at it but can sometimes see the beauty of math, I do have some advice.
I know that the times in my education when I felt most disconnected, disinterested and even disgruntled with math were the times when I felt that I was being told to learn things without being helped or allowed to understand their beauty or utility, especially when I felt I wasn't being given the time to do so. I hated the times when I felt that I was being made to learn without regards for my curiosity instead of through it or because of it.
But the times that I felt the most uplifted and filled with wonder were when I was taught something that brought down barriers, that allowed me to solve problems I'd have written off as unsolvable before. For example, learning long division after spending so long dealing with remainders.
Also uplifting are times when the amazing structure and endless order of math revealed itself to me. Playing with fractions and finding the different relationships between the numbers chosen and the fraction's value as a decimal--for example, seeing the way that the decimal forms of a series like (2/1, 3/2, 4/3, 5/6, etc. 'walk' their way downwards and trying to find the rules that governed such things.
Word problems can be engaging, but they should be something that makes the student care more about the answer, not just something that gives a flimsy and petty pretext for why the math needs to be done. The latter can actually make doing the problem seem less worthwhile. Less "A train leaves Detroit..." and more "Can this man get to the airport to stop the woman he loves leaving forever?!" Less "How much fence do you need for this garden" and more "What's the largest enclosure this tiger can have at this zoo without being able to escape with a running jump?"
So the times I loved were when I was being shown not just the what, when and where of numbers, but the why and the how. There's no need to include God in the "who" position, just try to let the students see the beautiful patterns in the numbers.
It is a fundamental question. Let me to add some brief additional points to other useful answers.
1. How can I add spiritual aspects to my mathematics?
Spirituality of your mathematics comes from your meta-mathematics. By meta-mathematics I mean all parts of human knowledge which discuss about mathematics including logic, philosophy of math, etc. It is a very important point. Your meta-mathematics determines the interpretation of the mathematical theorems which you teach. In fact, one can talk about all beautiful theorems of mathematics and all religious mathematicians of the history without any spiritual impact on his students when he doesn't use a spiritual meta-mathematics. Simply because beauty of a subject/object cannot lead one to any spiritual experience necessarily. Can looking to a beautiful flower convince somebody that there are some spiritual beings? Not necessarily. The same situation happens when one tries to increase the spiritual aspects of his teaching just by showing the beauty of mathematical theorems to his student. Also in the case of talking about faithful mathematicians there are many counterexamples for each religious person. For example in logic if you mention Cantor and Godel as religious logicians another one can mention Russell and Turing as counterexamples. All of these persons are great logicians with brilliant deep works. Thus the technical theorems of logic/mathematics cannot lead one to believe in God or any other immaterial existence necessarily. You need endow your mathematics with an appropriate philosophy to conclude such conclusions. If you choose a suitable spiritual philosophy of math as the standard philosophy of your courses then all of your lectures, theorems, proofs, discussions and activities seem really spiritual and leads your students to a spiritual experience.
2. What is the best meta-mathematics for a spiritual teaching?
Certainly, Platonism is a good option because:
- As a possible consequence of Plato's travels to the Eastern countries including Jerusalem (and maybe Persia) and meeting with Rabbis and Zoroastrian clergies his philosophical point of view is designed based on a pyramid. Plato's philosophical system implicitly leads one to a unique point on the top of this pyramid (Plato's God).
- Platonism suggests a clear pattern for uplifting. In Plato's point of view, the role of mathematics in one's spiritual travel from earth to the ideal world of ideas (Plato's heaven) is very important. Mathematics is a bridge in the route and maybe the last one of them in the modern age.
- Based on Platonic point of view, mathematical objects and theorems are discoveries not inventions. For those who believe in Platonism, each new theorem is a fact about the invisible world of mathematical objects.
- In Platonism, doing mathematics is a kind of praying (or meditation) which helps one to understand the absolute truth of the world.
Remark 1. The ideas of Pythagoras about the interactions of mathematics and world are related. Also the philosophy and theosophy of Plotinus and other philosophers of neoplatonism could be related and very useful.
3. How should I use the spiritual meta-mathematics in my teaching?
In one word, implicitly. Generally, there is no need to talk about Socrates, Plato, Pythagoras, Plutinus, Cantor, Godel or any other philosopher or mathematician in your teaching. (In special cases, for example when you are teaching special courses like logic or set theory, it is relevant and useful to refer to life and belief of some persons directly but it is irrelevant in usual math courses like calculus and linear algebra). What you need is just a Platonic approach and explaining his ideas about mathematics and mathematical objects in your implicit comments. Just as an example when you are talking about a particular mathematical object/theorem you can use a sentence in the following form:
In 1963, Paul Cohen discovered the Forcing method and a new model of ZFC and finally found the solution of one of the most important questions of 20th century.
Compare with this one:
In 1963, Paul Cohen constructed a new model of ZFC by his Forcing method and solved one of the most important questions of 20th century.
Asking questions instead of expressing a sentence explicitly could be another useful method to lead students to build their own spiritual world. For example you can say:
What phenomenon does Cantor's theorem on existence of more than one infinity, describe?
Compare with this one:
Cantor's theorem on existence of more than one infinity says there are some infinite and immaterial worlds beyond our finite materialistic universe.
You can find more useful techniques to make your teaching as mysterious as possible. Let students be free to have their ideas and imaginations.
Remark 2. Note that spirituality is an internal experience of each person. You cannot and should not force them to believe on existence of immaterial objects including God, Gabriel, $\pi$, $\aleph_1$, etc. What you can do is just motivating them to explore inside their minds and hearts to find these beings and unfold the invisible objects of the world.
Remark 3. Note that what you want to do is an ideological teaching in the sense of what I described in my post "Ideological Teaching in Logic Courses". This is not easy. You should be aware of possible negative and inverse impacts of this approach. Detect the feedback of your audiences continuously.
4. Do some fields of mathematics have more potential to be interpreted spiritually?
Yes, of course. The nature of some parts of mathematics is very adequate for spiritual interpretations. You can use this potential in your teaching by leading your students to these realms. For example in set theory you don't need to do too much effort to bring spirituality in your teaching. Almost all textbooks are full of the comments with explicit or implicit references to religious and Platonic beliefs. Here are some of the quotations from usual textbooks of set theory.
- ..., Kunen's result (Kunen inconsistency theorem) can best be viewed as an ultimate limitation imposed by the Axiom of Choice on the extent of reflection possible in the universe. $ZFC$ rallies at last to force a veritable Gotterdammerung for large cardinals! [Akihiro Kanamori, The Higher Infinite, Page 324]
- ...,Cantor came to view the finite and the transfinite as all of a piece, similarly comprehendable within mathematics, and delimited by what he termed the absolute which he associated mathematically with the class of all ordinals and metaphysically with God. [Akihiro Kanamori, The Higher Infinite, Page XII]
- There are more things in heaven and Earth Horatio, than are dreamt of in your philosophy - Hamlet, I.v. 166-7. [Akihiro Kanamori, Menachem Magidor, The Evolution of Large Cardinal Axioms in Set Theory, Page 1]
- ..., Someone once said to me, doing set theory with a universal set is like believing in God. [Thomas Forster, Set theory with a Universal Set: Exploring an Untyped Universe, Page 11]
- ..., To use a medical metaphor set membership presents as an allegory of predication but it is also an allegory in more obscure ways, for example of the dominance relations of the numerous conceptual hierarchies that people have dreamed up from time to time, be they of language levels or levels of existence, (humans, ..., angels,...,God). [Thomas Forster, Set theory with a Universal Set: Exploring an Untyped Universe, Page 11]
- ..., People living in $M$ cannot construct a $G$ which is $\mathbb{P}$ - generic over $M$. They may believe on faith that there exists a being to whom their universe, $M$, is countable. Such a being will have a generic $G$ and an $f_{G}= \bigcup G$. Then people in $M$ don't know what $G$ and $f_{G}$ are but they have names for them, $\Gamma$ and $\Phi$. They may also read the preceding few paragraphs and thus figure out certain properties of $G$ and $f_{G}$;...More generally, they can construct a forcing language, where a sentence $\psi$ of the forcing language uses the names in $M^{\mathbb{P}}$ to assert something about $M[G]$. [Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 193]
- Remark 4. Note that what Kunen describes in the above paragraph as a forcing language for the people living in $M$ is very similar to what we call religion in our world. Precisely a religion consists from sentences about names of some beings (such as God, Gabriel, Satan, ...) which some people of our world believe on faith that they exist but they don't know precisely what these beings are. However people can describe some of properties and relations of these creatures using their names such as: "Satan is the enemy of God". So religion and science are both a collection of sentences in the language of people living in our world with a difference that using names of generic objects which don't belong to our world are allowed in religion but in science every sentence must use names of real objects which are members of our world like any materialistic creature.
- ..., Philosophical positions on the foundations of mathematics have more marked impact on set theory than anywhere else in mathematics; and the reader should be well aware of the prejudices of the author he is reading. I have written this book from an uncompromisingly realist or Platonist position; that is, I have taken the viewpoint that in some sense sets do exist, as objects to be studied, and that set theory is just as much about fixed objects as is number theory. ..., It seems very difficult to me to give any reason for the study of large cardinals without taking a viewpoint of this sort. " [Frank Drake, An Introduction to Large Cardinals, Page VIII]
- ..., The Platonists of $2100$ may know whether $CH$ is true, but neither $CH$ nor $\neg CH$ will ever be a formal theorem of $ZFC$. Likewise, many Platonists of today believe that inaccessible cardinals exist, even though the statement that they exist is not a formal theorem of $ZFC$. [Keneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 11]
- ..., So our results are actually relative consistency results - i.e., predicated upon the assumption that $ZF^{-}$ is consistent. These relative consistency results will be accomplished by completely finitistic means, whereas the consistency of $ZF^{-}$ will remain either an open question or an article of faith, depending upon one's philosophy. [Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, Page 111]
- Remark 5. You can find more information about the religious backgrounds of set theory in these books:
- (1) Amir Aczel, The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind.
- (2) Loren Graham, Jean-Michel Kantor, Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity.
5. How can I prove existence of spiritual beings to my students?
You don't need to prove any thing! The faith begins when the proof fails! In fact the scientific and mathematical proofs as finite sequences of valid sentences based on a reasonable set of axioms, are too short to reach the realm of transcendental objects. The same situation happens when we want to decide on truth or falsity of Continuum Hypothesis ($CH$) or existence of large cardinals. They are independent from our set of axioms ($ZFC$). Set theorists believe on faith in one of $CH$ or $\neg CH$ based on their personal intuition about the Cantor's heaven. Of course this faith has an important influence on their mathematical approaches and mathematical life too. Let me to explain more using the following pseudo-logical terminology:
- (Soundness) One should believe on the truth of every provable subject.
- (Smallness) Scientific and mathematical proofs are too few and too short.
- (Largeness) Transcendental subjects and objects are too many and too far from our intuitive set of mathematical and physical axioms.
- (Incompleteness) Most Transcendental subjects and objects (e.g. Large cardinals, Angels, God, etc.) lie beyond borders of scientific and mathematical proofs.
- (Faithfulness) The only way to decide on existence of the transcendental objects and subjects is by intuition and faith.
Remark 6. Note that independent mathematical subjects like Continuum Hypothesis and existence of large cardinals are not meaningless, useless or ignorable. They are just undecidable. The same situation happens about the other transcendental subjects like angels and God.
In other words, in order to lead your students to spiritual subjects you just need to motivate them to think about these matters and try to talk about your (and other mathematicians) faiths.
6. Can I use formal language of mathematics to increase the spirituality of my teaching?
Yes. Beside their applications, mathematical notations and names could carry implicit meanings. Let me to explain by an example. In set theory we use several names and symbols which have religious meanings too. For example: Hebrew letters, cardinals, absolute, hierarchy, Cantor, etc. Also one can enlarge this list by some funny creative tricks. e.g.
You can use local notations and names of each field of mathematics effectively. This depends on your creativity.
7. Is there any special spiritual aspect in Linear Algebra?
Yes. Use the notion of dimension. The spiritual beings live in the spiritual worlds. These worlds are higher than our materialistic universe. What is the meaning of a higher world and a being of a higher type of existence? In some sense one can interpret it as a world with higher dimensions. The linear algebra course is the first place which students learn about the notions of space, sub-space and dimension formally. Try to relate these notions to religious concepts. Also you can compare the power of a three dimensional being (like us) in the world of two dimensional creatures (e.g. those who live in a sheet!) with power of a four dimensional being in our three dimensional world. For example:
- A human can put his finger somewhere on a sheet. The two dimensional inhabitants of the sheet think some new creature appeared in their world (the two dimensional intersection of finger and sheet). They come to see it! Then the human can withdraw his finger and put it somewhere else on the sheet far from the first place. They see that the strange creature disappeared suddenly and then appeared somewhere else. They may think this creature is traveling in their world like a ghost!
I like this one,for promoting a sense of awe and wonder..and inquiry: there is a 'gap' in the graph of the function - obviously at $n=1$, of $y = \big(\frac{n}{n-1}\big)^n$... since it is a gap, it must be able to be 'bridged' you would think, but since it also is a gap of zero dimensions it can't be: it both can and can't 'exist' simultaneously.. it's 'there'.. but when we try to measure it in physical terms.. it's not!
Well,
There are the perfect numbers like 6 and 28 etc. 6 was actualy called "Number of God". Because its a perfect number where the sum of there dividers always ends up to be the number himself. 1 + 2 + 3 = 6 and God created the world in 6 days !!
28 is also a perfect number (1+2+4+7+14=28) and was always be refered to as the Moon-Cicle which could only come from god.
Humans are known for there wild interpretations :-)
Funny though, I was able to remember this because of the wild interpretation !
