How would you apply this concept to a ring species in biology? i.e. one in which A can breed with B, B can breed with C, but C cannot breed with A.Mathematically, canonical representatives are given by a function f: X → X such that f(a) = f(b) if and only if a ~ b.
The definition of the quotient set works even in situations where ~ does not have canonical representatives.
For the ancients/non-moderns, f(Lassie) = f(Rin Tin Tin) = Platonic ideal dog.
Can you explain this please.Mathematically, canonical representatives are given by a function f: X → X such that f(a) = f(b) if and only if a ~ b.
The definition of the quotient set works even in situations where ~ does not have canonical representatives.
Maybe you can explain, but it seems that either ~ specifies a merely arbitrarily grouped set of objects (i.e., RD’s claim: there are no natural kinds) or, absent a separate Platonic ideal, each member of the set is necessarily a canonical representative.For the ancients/non-moderns, f(Lassie) = f(Rin Tin Tin) = Platonic ideal dog.
In modern set theory, one forms a set from a description by invoking the appropriate axioms that guarantee the existence of a set having that description. In this particular case (i.e., quotient set construction), this involves the axiom of separation/restricted comprehension applied to the power set axiom. See here for the axioms of set theory.So you claim, but how does this answer my question: how does modern mathematics make it possible to form (not just to describe - to form) equivalence classes without invoking some kind of canonical representative (such as a quiddity)?
Given a system of canonical representatives, we can define f(x) to be the canonical representative of x.Can you explain this please.
We must have f(Lassie) = f(Rin Tin Tin), so it is impossible for both Lassie and Rin Tin Tin to be canonical representatives.Maybe you can explain, but it seems that either ~ specifies a merely arbitrarily grouped set of objects (i.e., RD’s claim: there are no natural kinds) or, absent a separate Platonic ideal, each member of the set is necessarily a canonical representative.
I found these definitions online, do you agree with them:Actually, no. You have water existing under one of its accidents, ice. Then, water existing under the accidental property of liquid. Then, water existing under the accidental property of ice again.
from here
Essence – it is the “whatness” of a thing, or rather the “quiddity” (Latin: Quid – what); answers the question “what is it?” – the essential parts of a material essence are form and matter.
Nature – nature and essence are almost synonymous save the fact that nature implies the proper operation of the thing
Substance: a subject of being, a proper recipient of being
Accidents: an articulation of the subject, which has its existence in the subject
If these definitions are correct, then surely an icicle is “a subject of being, a proper recipient of being”? So why would the icicle be a substance by that definition?Form – actualizes matter
Matter – receives its being from Form; cannot exist without Form
Essential Parts – Form and Matter are the “essential” parts of essence, because they provide the “generic intelligibility of material essence or quiddity.”
Thank you for your profound post. Unfortunately some of it went over my head. Are you basically saying that there may only be two substances, the physical and the spiritual, and that the transubstantiation changes the physical substance into a spiritual substance? (I don’t mean to put words in your mouth, sorry if I misunderstood. )The question of “substance” is an important one for all Roman Catholics, since we are uniquely committed to believe that at the consecration of the bread and wine, the substance of the bread and wine is transformed (replaced) by the substance of Jesus’ body and blood. Furthermore, the transformation of substances takes place without a transformation of the form and properties (accidentals) of the bread and wine. Also we know from the doctrine of the Trinity that God exists as three persons sharing the same substance. So, we can deduce that there are at least two substances existing in reality: material and spiritual. We can presume that the spiritual substance is unique to the Trinity and therefore infinite in both extent and divisibility.
So what do we say about the material substance of the bread and wine other than it is theologically different than the spiritual substance? First of all, modern science does not deal in the concept of “substance”, material or spiritual, because the philosophical principle to which science adheres is Logical Positivism. Logical Positivism emphasizes the meaning and use of language, but also defines the tenets on which modern science is based.
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Er… right… So can you explain how this answers my question (maybe you could use your dog example to illustrate how it works)?In modern set theory, one forms a set from a description by invoking the appropriate axioms that guarantee the existence of a set having that description. In this particular case (i.e., quotient set construction), this involves the axiom of separation/restricted comprehension applied to the power set axiom. See here for the axioms of set theory.
…such as?? (Note that the phrase “*Given *x, blah blah blah” usually indicates the possibility of a merely counterfactual claim.)Given a system of canonical representatives,
we can define f(x) to be the canonical representative of x.
Given a function f(), we can define the canonical representative of x to be f(x).
This conceptual equivalence establishes that f() is a valid definition of canonical representative.
We must have f(Lassie) = f(Rin Tin Tin), so it is impossible for both Lassie and Rin Tin Tin to be canonical representatives.
A ~ B if and only if A and B are the same kind of thing. I don’t see where any of my claims (1), (2), or (3) has anything to do with whether the kinds are natural or otherwise.
I fear I’m at the limit of what I’m able to explain about my idea. The quotient set construction is a such a staple of mathematical reasoning that I’m unable to see what your objections are. Anyway, I feel that my argument has benefited from this discussion, and I feel that my statement at the end of post 68 is an improvement from what I originally had.Er… right… So can you explain how this answers my question (maybe you could use your dog example to illustrate how it works)?
My objection is simple: I don’t see how you apply the abstract notion of quotient set (or just equivalence class) construction to the real problem of constructing the set of all dogs (for example). And you can’t explain how this is done?I fear I’m at the limit of what I’m able to explain about my idea. [With due respect, I fear that too.] The quotient set construction is a such a staple of mathematical reasoning that I’m unable to see what your objections are. Anyway, I feel that my argument has benefited from this discussion, and I feel that my statement at the end of post 68 is an improvement from what I originally had.
In general, I agree with them. But, as to specifics, it has to be remembered that “substance” has many definitions. For the Scholastics, though, the definition was specific and philosophical. Remember, “substance” is defined above as a subject of being, not the subject of being.I found these definitions online, do you agree with them:
"Essence – it is the “whatness” of a thing, or rather the “quiddity” (Latin: Quid – what); answers the question “what is it?” – the essential parts of a material essence are form and matter.
Nature – nature and essence are almost synonymous save the fact that nature implies the proper operation of the thing
Substance: a subject of being, a proper recipient of being
Accidents: an articulation of the subject, which has its existence in the subject
Form – actualizes matter
Matter – receives its being from Form; cannot exist without Form
Essential Parts – Form and Matter are the “essential” parts of essence, because they provide the “generic intelligibility of material essence or quiddity.”
If these definitions are correct, then surely an icicle is a “subject of being, a proper recipient of being”? So why would the icicle be a substance by that definition?
Not anymore than I already have. Sorry.My objection is simple: I don’t see how you apply the abstract notion of quotient set (or just equivalence class) construction to the real problem of constructing the set of all dogs (for example). And you can’t explain how this is done?
Neil,Thank you for your profound post. Unfortunately some of it went over my head. Are you basically saying that there may only be two substances, the physical and the spiritual, and that the transubstantiation changes the physical substance into a spiritual substance? (I don’t mean to put words in your mouth, sorry if I misunderstood. )
Since modern physics pretty much disregards the notion of “substance” and focuses only on physical properties (accidents). In fact many people would deny that substance is real.Consider a metal bowl. Its accidents include its diameter, its mass, its color, shape etc. Its substance is “bowl”. Now, imaging a craftsperson who uses a hammer to flatten the bowl into a plate. Now the metal has a different shape (one of the accidents), and a different substance. It’s new substance is “plate”.
Speaking as a physicist, I don’t agree with your premise that modern physics pretty much disregards the notion of substance; it may term the essential character of an object by different terms (particularly if describing it quantum mechanically). And if you go back to Aristotle’s “Physics” he treats the example you gave (quoted from the internet): * So the ingredients Aristotle insists on are: an underlying subject, a form (i.e., a positive property) and a lack (or privation) of that form. Aristotle’s examples illustrate these ingredients:*
*In case (a), the subject is man, the form is musical and the privation is unmusical. In case (b), the subject is bronze, the form is statue and the privation is shapeless. The subject—the man, or the bronze—persists through the change. Of the other terms involved, the earlier ones (unmusicality, shapelessness) cease to exist, while the later ones (musicality, the statue) come into existence.Code:*1. A man who was unmusical becomes musical.* *2. Some bronze (which was shapeless) becomes a statue. *
These were cases of coming into being (generation), since lacks or privations were replaced by forms. Ceasing to be (destruction) occurs when a form is replaced by a privation—when matter is deprived of form. This would happen, for example, when a statue is melted down into a shapeless pool of bronze. The bronze persists, but the statue has ceased to exist.* Your example is a change of form, not substance.