Objectivism: Intensionality Matters.

Recently myself and a number of others have been engaging online with a group of “Objectivists”, of the Ayn Rand kind, and to say their misunderstandings of philosophical concepts are profound, would be a bit of an understatement. They seem to consider themselves some type of “deep thinkers”, but generally only tend to only present a very basal and superficial understanding of the complex topics that they try to discuss online. They also tend to be centered around actor, and self-identified Objectivist, Mark Pellegrino (Lucifer from Supernatural) who was an amazing guest on my YouTube Channel, and who can certainly carry his own weight on many of these topics…even if I fundamentally disagree with some of his views. But I’m hoping this blog post can elevate the discussion and elucidate some of the key concepts that we have been discussing back and forth. (Some even argued you can’t start a sentence with “But”, but you know me, I’m such a rebel!)

As I have said before, more times than I could even remember, Twitter is a cesspool and not really the place the be able to suss out nuances in many of the things this group bring up, and they for some reason seem to have notable aversion to “jargon”, any potential for productive dialog is irrevocably lost by being bogged down in petty semantics or trivial points of little to no relevance to the fundamental philosophical concepts at hand. It is difficult enough to explain very complex concepts, even more so on Twitter, and even more so when they use a vernacular of a private language all their own and don’t understand philosophical terminology except to throw philosophical words out there, almost seemingly randomly as if they are some type of defeater to an argument. Also as I have said before “Your misunderstandings are not refutations to my arguments”.

One of the sticking points to this group of Objectivists is the concept of .999… = 1. Now, don’t worry I’m not going to get into mathematically why that relationship holds or prove it here. (Take a second to relax by taking a deep breath of relief if needs be here!). However, I am going to attempt to discuss some of the conceptual issues I see in relationship to that group of Objectivists in regards to their balking at such a relationship.

Their argument is apparently something along the lines of “.999…” and “1” are different objects and therefore not the same, such that in that saying something like “A is A” is not the same as saying “A=A”. Logically of course that if you were to symbolize “A is A” it would be represented as the equality “A = A”. Using philosophical terminology, that they vigorously avoid using, which makes trying to ascertain what they are trying to express vastly more difficult, I would say they arguing that the signifiers (the actual string of numbers, word, symbol you are looking at on the page) are different, thus not equal. Whether or not they recognize what is being referenced to both “.999…” and “1” is the same exact referent, in this case being the same exact point or value on the real number line, I do not know, as they are very vague and tend to prevaricate substantially in answering those types of questions. In either case, what is being signified is the concept (or idea) that the symbolism or word evokes, which is that “.999…” and “1” both convey the concept of the midpoint equidistant on the interval of [0,2].

My general impression of their idea of “deep thinking” is somewhat of a bastardized concept of “intensionality”, in that while mathematically .999… and 1 are equal, there is an intensional difference to be had between “.999…” and “1” given an instensional statement or proposition. They are arguing that “A is A” is an identity, in that no matter what I instantiate “A” with, the equivalence relationship holds and that it is conveying that an object is itself. While the equivalence of A=A holds mathematically, to them, expressing a mathematical equality or equivalency does not expressly convey that an object is itself, merely that they are of the same value. To them, I guess, the metaphysical aspect of ontologically “an object is itself” gets lost in translation or something when you try to express “A is A” mathematically, even though mathematics is a type of metaphysics.¹

Leibniz’s law or the identity of indiscerniblity states that if two objects have the the exact same properties, then they are the same object or more basically the law of identity in that ∀A(A=A). For any A or “All of A”, the equation A=A holds true. More formally we can express that as:

∀x∀y[∀F(Fx ↔ Fy) → x=y]

If x=”Steve McRae” and “y”=”myself” (referring explicitly to me here) then since Steve McRae and myself have all the exact same properties (F), then we are in fact the same object. Objectivist in that group denied two objects can have the same exact properties and thus be the same object, but I think failed to recognize what was really being expressed by the concept of the identity of indiscernibles. “Steve McRae” and “myself” are two different objects, as they are intentionally different, but extentionally equal in that they both refer to the same object (person) writing this blog post.

But given the maximum principle of charity here, let’s perhaps try to suss out more their apparent argument by first explaining “inten**s**ionality” and “inten**t**ionality”:

In philosophy, words like “believe”, “think”, “love”, have intentionality as they are signifiers that are mental states that are some representation towards something. “To believe” or “to think” are ‘intentional verbs” as you have to believe *in something*, or believe something to be true (or false), or to think *about something*. For example you can have a thought about something, while physical objects can not…they merely have extensionality in space and time, but they themselves have no intentional states. Intensional is any property that is connoted by some definition, to be both necessary and sufficient, about a referent object which are construed to be “intensional objects”. One of the things that came up in discussions with this group was a claim that humans being apes is a metaphysical necessity. I am going to avoid going into the severely bad, and hopefully obvious, athletic modality issues here, as if that claim were true then in every possible world, humans must be apes, even though “human” being apes is categorical and contingent.

However, the intension of the word “human”connotes a being which specific properties if what we have arbitrarily (this was another point of contention by them not accepting the properties of being human are of a human construction of arbitrary conditions) we assigned to be what we call the word “human”.² For example, “human” connotes an object with the properties of being multicellular, chemoheterotrophic, bipedal, living, animal, eukaryotic etc. If I could list every property of what is constitutive of we mean by being “human”, I would have what is called a object called a “comprehension”, which is the totality of intensions that give meaning to what we have arbitrarily prescribed to meet the conditions to call an object “human”. If we could list these properties we would have an “extensional” set which also describes what it is to be referred to by the word “human”.³

For example:

Set A=”All of the States of America”

Set A is intensional in that it connotes the extensional set B={California, Alabama, New York, Florida, …} if Set B here contained all 50 states. Both sets describe the same exact thing. However, they are clearly not the same literal set.

Set A=”All of the States of America”

Set B={California, Alabama, New York, Florida, …}

Both sets contain the same number of elements, and but are not intensionally the same.

Mathematical functions would be similar. If we had the algorithms (using from Wiki for a simple example):

To find f(n): first add 5 to n, then multiply by 2

To find g(n): first multiply by 2, then add 10

Both f(n) and g(n) are extensionally the same as for whatever value you put in for n you will get the same resultant. They are however intentionally different as they don’t refer to the same process for determining n.

These intentional differences are such that you can’t always directly exchange f(n) for g(n) in an intensional proposition. Intensional propositions are such that in at least one case replacing coextensive expressions will change the truth value of the proposition. A canonical way of explaining this is to use the position is given by:

p=”Lois Lane believes that superman can fly” Is true

If however Lois Lane doesn’t know Clark Kent is Superman, even if coreferential, that if you substitute “Clark Kent” for “Superman” into the proposition p:

p=”Lois Lane believes Clark Kent can fly” it changes the truth value from T to F, as she doesn’t know Clark Kent is Superman.

I can believe Superman can fly, that is intentional (with a t) in that I am thinking (having mental content) about something, and intensional (with an s) in that thinking about Superman can fly doesn’t mean I am thinking that Clark Kent can fly. If Superman loves Lois Lane that would be intentional (with a t) as Superman has an intentional verb (loves) with respect to some object x (Lois Lane).

If this was quantified it would be as:

Ǝx(Superman loves x)

Which means there exists an x that Superman loves, with x here being Lois Lane. The existential qualifier “Ǝ” here does not imply existence. It does not mean that Superman in fact exists, or has “actually” has intentional states. This was another conceptual issue in the group of Objectivists, in that they balked at using “Superman” in a proposition merely because Superman is a fictitious object, which no “real world” referent that exists. Objectivists apparently believe that there is no object in reality that some concept points to, then that concept is somehow “in error” of some kind, almost like some type of radical form of correspondence theory of truth.

So going back to the Objectivists argument against .999… = 1, while some accepting it is mathematically equal, but deny it being being true due to “metaphysical” reason, or “A is A”, they are trivially, and rather clumsily and ineffectually, trying to argue that “.999…” and “1” are different symbols and, while co-referential, are intentionally different.

For example:

p=”.999… is a repeating decimal” is true. However, if you replace it with the equality of “1” you have:

p=”1 is a repeating decimal” which is false.

The truth value of p has changed by this substitution which means the proposition is referentially “opaque”, such that even though .999… and 1 refer to the same mathematical value, you can not always merely substitute one in for the other in a given intensional proposition. This to me however is a very trivially true argument at best and does not outweigh the claim that .999… = 1 is false as so many of them make, and many of them do argue mathematically that .999… < 1, which wrong prima facie regardless of what petty and trivial metaphysical arguments they try to bring to the table.

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¹ We don’t do empirical testing on mathematics, we don’t do empirical checking on premises, axioms or postulates in maths, and as Russell argued we don’t even start out when evaluating if an initial assertion or premise in pure mathematics is true or not, but we proceed on the premise of* if it was true. *

“It is essential **not **to discuss whether the first proposition is really true”” – Bertrand Russell

³ Humans being apes is arbitrary as part of an extensional set. Monophyletically we are apes, so ape would be in an extensional set of properties for “human”, but paraphyletically we are not apes and would not be in said set.

William PiiI would agree that part of the issue is multiple representations for the same object. This is the same reason that I think people have issues with the rational numbers, let alone the real numbers. After all, the fraction “one-half” has multiple distinct, but equivalent, representations: 1/2, 2/4, 3/6, etc. These are all different ways of writing, and describing the same concept: the midpoint between 0 and 1. The analogy I give to students is that “1/2” is like me wearing a suit and tie, but “2/4” is like me wearing comfy sweatpants. They are the same thing, just dressed up differently. Considering Leopold Kronecker is credited as saying, “God created the integers; all the rest is the work of Man,” I am inclined to think that representation problems like this are well-known and persistent.

However, one can prove that the representations of “1/2” and “2/4” are equivalent. The construction of the rational numbers has an equivalence relation embedded within it, specifically p/q = r/s if and only if rq=ps in the integers. For a formal treatment, please see “localization” of a ring in Lang or Dummit/Foote.

Interestingly, the integers also are constructed with an equivalence relation as well when using Grothendieck’s group construction, which is in many K-theory books. Please see Weibel’s “The K-Book” for a formal treatment. This construction is one way of constructing the integers from the whole numbers (i.e. natural numbers with 0), but due to the properties of the whole numbers, there is canonical form that all integers take. Namely, every integer can be written as a-0 or 0-a, depending on sign. Sadly, the same cannot be said for the rational numbers, and the best that can be done is to find “lowest terms”, i.e. when the numerator and denominator are coprime.

The same can be said of “1” and “0.99…”, though the equivalence relation is messier to describe as it involves limits, Dedekind cuts, Cauchy sequences, or other equivalent analytic criterion for cofinality. Please see your favorite analysis book, such as Davidson/Donsig, or maybe an order theory book like “Ordered Sets” by Bernd Schroeder. However, as with “1/2” and “2/4”, one can prove that “1” and “0.99…” are equal from their construction.

Side note: The fact that 0.999…=1 is true demonstrates that limits do not preserve strict inequalities. Specifically, the partial sums (i.e. 0.9, 0.99, 0.999) are all strictly less than 1, but the limit of this sequence is dead-on equal to 1. Perhaps this is another stumbling block for readers/viewers?