Intensional philosophy

16 September, 2008 at 9:49 pm (philosophy) ()

This is a pretty significant paradigm shift for me. It may or may not be of personal interest to you.

Of intensionality and extensionality

First, an example from the field of set theory. If I take some set, like the set of people who read this blog, I can mean at least two things with it. The first meaning is the set of people who currently are reading this blog, or are currently readers of this blog in some other sense. I could in theory name them and list them, like so: {Tommi, ksym, Phil, Fred, …}. This is the extension of the set. On the other hand, I can mean any people who are readers now or who might be readers some day or who could potentially be readers. In this case the actual list would be largely irrelevant; the way the set is defined is what matters, not the contents it may or may not have.

A more philosophically interesting example: I have two tables in my room/apartment. Assume, for a moment, some ontological theory that ascribes existence to single material objects only, not classes of them (like “tables”). Now, consider, can I meaningfully refer to the two tables as, well, two tables, as I have been doing here? For a mathematician, the answer is “of course”. If I have two entities, I can certainly define a set to which those two belong (assuming the entities are not built so as to resist this). That: The extension of the set is what matters, the intension is irrelevant and in this case not even distinct. But a philosopher would think about the intension: By what means can I refer to the two tables, if they are ontologically only arbitrary physical objects? Certainly not as I have been doing, because given the ontology here, “table” is not very meaningful (at least obviously).

So, in closing: In math, intensionality is a means to an end or a red herring; the extensional is what matters. In philosophy, the intensional is the interesting parts, and the extensional may or may not matter. Philosophy is about the why of the world, or segments thereof, while math (and physics and other hard sciences) about the what and how.

As a disclaimer, the parts about philosophy apply most to ontology and metaphysics and ethics by Kant, and maybe to other stuff I am less familiar with. Also: Almost all distinctions are fuzzy around the edges.

What this means to me

This realisation has been fundamental in that now I no longer see significant parts of philosophy as trivial, which is useful for being motivated to actually study it.

Furthermore, I will need to re-evaluate my interest in philosophy. It requires learning a new way to think, which is generally fun and useful. Can I learn to think philosophically? Maybe. Do I want to? Maybe.

In case I am utterly wrong

In case I am misrepresenting large portions of philosophy or mathematics, please do inform me. I will presumably argue against such a claim, which will force me to sharpen my thoughts on the subject, which is useful. Or I may admit to being wrong.

In case you know something about the subject matter

I would be interested in any literature concerning this subject. Very interested. So, if you know any, I would much appreciate you sharing the information.



  1. the_blunderbuss said,

    Hey Tommi, I’m glad to see you again =)

    I’m going to need further clarification. First because I’m very insecure about discussing this sorts of topics in a language that’s not my own (topics that appear very interesting to me.) Second because I fear I might have misunderstood you. Regardless, these are my two cents:

    We experience and perceive the world (whatever that might be, the term is sufficiently evocative I think) in a symbolic way. There is no way for us to experience every entity as a completely individual and unique thing without experiencing them as entities. By the same token (and as an answer to H.P. Lovecraft) we can’t experience the truly unspeakable. We would most likely not experience it altogether (regardless of the actual senses involved in the process, the experience requires more than mechanical reaction to stimuli) or we would (much like that old chap though) go terribly insane.

    What I meant when I said that we experience and perceive the world in a symbolic way, was that the reality that we dwell in is a reality of meaning. And meaning lives only by the merit of contrast. Contrast requires commensurability (here as ‘the capacity for two or more things to be measured with a single scale’) and that brings us back to ‘tables’.

    I hope I could get the point across. A pleasure to read you as always.

  2. d7 said,

    As it happens, this is right in my line of philosophical interest. My studies have been somewhat casual, though, so I don’t have a good mental index of sources I could cite. I’ll see what I can dig up on intensionality in my library.

    I can’t comment on the mathematician’s take on intension, but you have the philosophical take right (presuming the ontology you gave to begin with). These sorts of problems about tables and categories and meaning (as mentioned by the blunderbuss) are part of why I subscribe to a layered ontology. It allows for the tables to be discrete, individual, and incomparable objects on one layer, while their both being “tables” is allowed for on the layer of meaning in which we conscious beings operate. I have some ideas about how the layers can interrelate, but they’re difficult to explain in a way that makes any sense.

    I don’t have the backing of any philosophers on that particular ontology, but it’s the only way I’ve found to reconcile the problems presented by determinism, apparent free will, questions about the nature of mind, the inherent functionalism of scientific description, and other diverse areas of philosophy.

  3. Syrsuro said,

    This issue is also of direct relevance to Systematics (the science of classifying organisms).

    At one level, systematics works by taking groups of organisms and deducing a set from them (“Ants”), defining the characteristics of that set and then defining all organisms that share those characteristics – including those not found yet – as belonging to that set. But systematics only ‘works’ if the set itself has meaning – i.e. if the organisms assigned to that set share characteristics (ideally shared ancestry) that go beyond the characteristics used to assign them to the group.

    Thus in systematics, unlike both math and philosophy, both the extension and intension of the set matter.


  4. Callan said,

    “So, in closing: In math, intensionality is a means to an end or a red herring; the extensional is what matters. In philosophy, the intensional is the interesting parts, and the extensional may or may not matter.”
    I dunno, but I think your working from the assumption that something matters, then trying to locate what it is. Why not try looking at it like a large equation, where there is nothing that matters at all, just numbers with mechanical relationships between them?

    From my perspective, you don’t find meaning in the equation that is life. You look at it, then decide what is meaningful.

  5. Tommi said,

    Hello Fred.

    You are engaging with my example ontology, right? I don’t really support it in any meaningful way, and trying to defend it is beyond my current ability. If you to read more about related matters, books that use the word “ontology” are a good guess. Or I might formulate an ontology I can actually defend. Or ontologies.

    Hello d7.

    That looks very interesting. I’d be interested in further discourse on the subject matter or further reading on layered ontology.

    Greetings Carl.

    Your field seems to indeed be one where both aspects of sets are focused upon. (Both are used in math and philosophy, too, but one is more in the role of a tool and another the target of investigation).

    Hello Callan.

    I’m trying to understand philosophy in a way that makes it nontrivial, mostly. In this case, what matters is specific to a given field.

    As for life in general I presume people make their own meaning, so we agree on that front. It is simply not overtly useful to trying to understand philosophy as a field of study.

  6. Callan said,

    Well, I don’t think you can understand it as just a field of study.

    Lets say philosophy is about trying to understand the meaning of the universe. (probably a bad way to put it, but humour me)

    If you could say you fully know that field of study entirely, then you would know the meaning of the universe.

    You can’t know this as just a field of study. The context of philosophy is dealing with life – it’s a lifestyle more than a field of study – once you read it in context with dealing with life in general, then it’ll light up for you, in various ways.

  7. opusinsania said,

    “But a philosopher –” you are actually describing is weird one in that he happens to be a reductionist, a view held mostly by scientists in the natural sciences, and not actually many philosophers. The problem of intension is easier to answer if one does not confine oneself to a ontology of pure physicalism. One example of an answer (or a beginning of an answer) could be the three-tiered ontology proposed by Popper ( that would probably define classes as objects of the third world. But ontology isn’t really my strong field, so this is mostly guessing.

    Your definition on the differences between philosophy and natural sciences (and math? this seems strange to me, considering that math is purely formal) seems a bit skewed, but informative nonetheless. A good perspective one might say, if not the only one.

    I also seem to disagree with Callan a bit: I think philosophy is and should be considered a field of study, although a very diverse one. This does not cover all of it, though, like existentialist writing by (e.g.) Sartre, but this can also be considered either in the context of the academic field or as a “life-philosophy”, which may be very enlightening also.

  8. Tommi said,


    1. Understanding something does not imply knowing it fully. I understand mathematics but do not know it fully. (No human does, I’d say.)

    2. I am interested in philosophy as a way to think and a field of study. This means that it will light up in different ways than it would for someone who embraces it as a way to live. I presume.

  9. Tommi said,

    Hey Opusinsania (the previous post was cross-posted);

    The ontological model is very much one that I do not support; it is just an example of the questions people thinking in different ways might ask.

    The very formality of mathematics is what makes it devalue the intensional parts of, well, whatever is studied.

    It is an axiom of set theory that two sets are equal if and only if they contain exactly the same elements/members. So, to use some pseudo-set-theoretic notation: {0, 1} = {the numbers signifying truth values in two-valud logic} = {the values an indicator function may have} = {the numbers used by base two system}. In terms of math all of those are exatcly the same sets (assuming the normal definitions). Are they really? That is a question for the philosopher. One can certainly construct a two-valued logic that uses 0 and 2 to mean false and true. Nothing anywhere to prevent it. If such logic is used, the relevant equalities above would no longer work.

    One analogy is this: In mathematics, things are, in a way, static. One does not create new circles or triangles or functions or graphs when doing mathematics; they are, in a way, already there (as their existence is a consequence of the axioms). The mathematician just gives specific ones specific names. This particularly means that a given set is static. Hence, every intension has a specific, static extension. This is not true of real life, where sets like {x | x is a person in my home} have their meaning changed as a function of time.

  10. Callan said,

    Fair enough, that’s valid. But if you don’t want to know it fully, how much do you want to know it? Do you decide how much? Does someone else decide? Is there some mash up, where someone else decides but after a certain point you’ll put your foot down and decide for yourself? Those questions come to mind.

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