Connectedness

I’m a student of the University of Colorado again. I’ve gone back to finish my bachelor’s in mathematics, which essentially involves fulfilling a bunch of core requirements. I’m going to start the discussion by mixing my experience of one class (religions of south Asia) with a concept from another class (connectedness from topology).

Last spring I took my (now ex-)':girlfriend on a trip to Hawaii. While we were there, we attended a weekend immersive class on Sanskrit. The class was very “new-agey” — we chanted, meditated, in addition to learning Devanagari (the Sanskrit/Hindi alphabet) and something about Indian religion. The ideas combined with the approach fascinated and inspired me. I have never been much of a religious person; the religious ideas I had heard of always sounded a bit naive and silly. But this new approach gave me a glimpse of another way of looking at the world: the words of the Bhagavad Gita played with the gods, using them half as entities, half as concepts. The philosophical ideas, language, and religion we studied were clearly inseparable, all connected and synthesized into a single world view. Further, this world view seemed to incorporate my objections to the naivety of western world views — emphasizing the duality in all things, focusing not so much on right and wrong but on purpose and spirit, using the malleability and metaphor of truth.

My curiosity whetted, I enrolled in a class about Hinduism at the university. So far it has been a disappointment. What drew me to these ideas in the first place was the connectedness and duality — the yin and yang, so to speak — I perceived in the world view. And we have started by drawing thick lines categorizing the different approaches to divinity. An especially potent event in bringing to my attention my disappointment with the class occurred during our discussion of Bhakti. The professor began to describe the philosophy of Bhakti: that connecting with the divine is about love and devotion, that the details of ritual are not as important as a true spiritual devotion to god. Immediately after this description, the professor wrote on the board BHAKTI RITUALS. Um, teacher, did you not feel that just now? How did you build your immunity to cognitive dissonance?

We have been categorizing, deconstructing, analyzing this beautiful philosophy as if engineers. After the class I suspect I will know many facts, but have no understanding. If I were to talk to a yogi, he will consider me no closer to understanding his spirituality than any other American out of the hat. This is disappointing, since I don’t consider myself to have learned something until I understand it. We have a Hindu temple here in Boulder; I hope to find a way to study there and use the class as a supplement.

But why I am really writing this post is to help me to grip a vague sense I felt as I was processing after the BHAKTI RITUALS class. I am in a topology class this semester, and we are learning set-theoretic point-set topology. The constructivist in me winces every few minutes, lamenting the non-computability of everything we are discussing. I think the same cognitive orientation is fueling my dissatisfaction with the Indian religions class and my taste for constructivism. Classical mathematics seeks to separate the world into true and false, existence and nonexistence, equal and inequal. The inclusion of the law of excluded middle as obvious is evidence of this, as is the surprise felt by the mathematical world over Gödel’s incompleteness theorem. “What? We can’t eventually separate everything into two categories?!”

If you ask a set theorist whether ℕ = ℚ, they will probably say they are not equal (although have equal cardinalities). If you ask a type theorist whether ℕ = ℚ they will say “huh?”. The question cannot be answered, for we must consider what it means to treat 1 : ℕ as a ℚ, and we don’t know how to do that — not without a function that shows how. Indeed, in constructivism we have to be careful when talking about real numbers, since the set of observations matters, i.e. it matters how we look at them. And for any reasonable construction of the reals, their connectedness falls out of the constructivism of the theory: we cannot separate them into two categories in any way. A set theorist can, and has to define himself into a more realistic world where he can’t using the mechanism of topology.

Mathematicians are probably getting upset at me or thinking I’m an idiot. This isn’t a mathematical post, it’s philosophical, thus my fuzzy intuitive discussions. If you have the desire to leave an emphatic corrective comment at this point, maybe take a step back and try to make out the landscape with me. I don’t consider any of this true, I’m just trying to get a feel for the philosophically general idea of connectedness, outside of a particular formal system. I have the impression that we can think of the world — the real one or the mathematical one — this way and it might lead to a more accurate, if less “clear-cut”, way of thinking.

The pure untyped lambda calculus is connected in the Scott topology. This fact has fascinated me since I heard of it, trivial though it might be. We are used to adding traditional totally disconnected types to the lambda calculus and pretending bottoms don’t exist. I have been curious about what it would look like if we embraced this connectedness and extended lambda calculus with connected concepts. They may play more nicely in a connected system. I still have not made any concrete progress on this idea, but it appeals to me as potentially beautiful and powerful. Maybe we are computing in an awkward way without realizing it.

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4 thoughts on “Connectedness

  1. Hi Luke, I’m a mathematician and I’m not upset at you and I don’t think you’re an idiot at all :)

    For ideas surrounding connectedness and topology in lambda calculus, you should check out Paul Taylor’s body of work on Abstract Stone Duality (http://www.paultaylor.eu/ASD/) if you haven’t already – it’ll be right up your alley.

    If you need a different motivation to study classical general topology, try the book by Steven Vickers: Topology via Logic. (And the follow-up paper: Topology via Constructive Logic).

    For constructive conceptions of topology, maybe you’ll enjoy work by Giovanni Sambin.

    Thanks for posting. Best,
    Ulrik

  2. I think there’s a deep seated discrepancy between those of us coming from the constructivist mindset (common among theoretical computer scientists, though not necessarily to computer scientists in general) and anyone coming from the classical mindset (the majority of mathematicians and, AFAICT, the near totality of philosophers). That is, deeper than is generally acknowledged.

    I find this unsettling because we’re definitely engaged in something which is very mathematical, very logical, and very philosophical, but all too often the mathematicians, logicians, and philosophers I run into are completely unwilling to analyze the classical assumptions they’re used to presuming. Whereas my constructivist peers are perfectly willing to accept that there are multiple perspectives, classicalism is but one of them. Ultimately this discrepancy means that constructivists cannot have discussions on their own terms with classicalists. Coming at the situation from an anthropological perspective, this balkanization serves to marginalize constructivists: either we can participate in the dominant classical dialogue, or we can be silent. This is an insidious attack on the constructivist programme and one that needs to be challenged vocally.

  3. wren ng thornton :
    I think there’s a deep seated discrepancy between those of us coming from the constructivist mindset (common among theoretical computer scientists, though not necessarily to computer scientists in general) and anyone coming from the classical mindset (the majority of mathematicians and, AFAICT, the near totality of philosophers). That is, deeper than is generally acknowledged.
    I find this unsettling because we’re definitely engaged in something which is very mathematical, very logical, and very philosophical, but all too often the mathematicians, logicians, and philosophers I run into are completely unwilling to analyze the classical assumptions they’re used to presuming. Whereas my constructivist peers are perfectly willing to accept that there are multiple perspectives, classicalism is but one of them. Ultimately this discrepancy means that constructivists cannot have discussions on their own terms with classicalists. Coming at the situation from an anthropological perspective, this balkanization serves to marginalize constructivists: either we can participate in the dominant classical dialogue, or we can be silent. This is an insidious attack on the constructivist programme and one that needs to be challenged vocally.

    I find that if you stick to treating logic as a tool and make sure that you always choose the right tool for the job you cannot ignore either classical or constructive logic. As soon as you start talking about the “real world” (and that includes computation) classical logic fails to adequately capture the totality of experiences and intuitions we have regarding “tangible” phenomena. I cannot imagine that a philosopher who is serious about using logic would be biased towards classical logic. Surely, for philosophical arguments, the law of excluded middle and its consequences should not be accepted without justification.

  4. I don’t think you’re an idiot, but I think that when you go to the temple, you may find that your university class is closer to traditional Hinduism than the retreat in Hawai‘i was. You could argue that traditional Hinduism has diverged from its true roots, and maybe you’d be right, but it happened a very long time ago. If you read sutras from 2000 years ago, they’re full of endless schemes of categorization.

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