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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 CoursesIdeological 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.

user230

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 pronounce the large cardinal symbol $$0^{\dagger}$$ as zero-cross not zero-dagger because cross is more peaceful than dagger!
• You can find a relation between ordinal and ordinance!
• You can rename ordinal trichotomy theorem as trinity theorem!

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!