Metaphysics Book X, Ch. I is probably a good place to start. How familiar are you with Aristotle's treatment of the "Problem of the One and the Many" and discussion of causes, principles, and measures? — Count Timothy von Icarus

Oh man, you're killing me. I'm an Aristotelian, through and through, and I've been meaning to talk to you, but it's just too much to take in at the moment, it's too much information.

You could also think of unit and measure as causes of number in that it does not seem that we would develop such concept if not for the fact that phenomenal awareness is full of numerically discrete entities that share a measure (and principle of unity). And it does not seem that this would be in phenomenal awareness unless it existed in the world (which our empirical investigations would tend to support), hence, unit and measure are causes of number in that sense as well.

Perhaps 1 is necessary, but not sufficient for 2. Just babble from my perspective.

Without 1, 2 could not exist, though the reverse doesn’t hold. Since it is because of the existence of 1, or one thing, that there can be 2, or two things, then the former can be said to be the cause of the latter. — Pretty

Here is my non-mathematician's understanding - All arithmetic comes back to counting. Each counting number has a name (1,2,3...1,000,000,001...). It is common to look at this in terms of sets. Each number represents a set with that many elements in it, e.g. 5 represents {x, x, x, x, x}. Addition is the same as the union of sets. 1 + 1 is equivalent to {x} U {x}, which is {x, x}, which is represented by 2. Does {x} U {x} cause {x, x}? Can there be {x, x} without {x}?

thanks for these references! I will totally use them to investigate further. Is one other way to put it that we should not confuse a part of a thing as a cause of it? Or should we be sure to be strict with this term “principle”?

Wouldn’t gravity be a perfect example of one? — Pretty

No, not at all. Ontological relations and gravity (and forces in general) are two very different things.

Wikipedia - Gravity

In physics, gravity is a fundamental interaction primarily observed as mutual attraction between all things that have mass.

SEP - Relations

Some philosophers are wary of admitting relations because they are difficult to locate. Glasgow is west of Edinburgh. This tells us something about the locations of these two cities. But where is the relation that holds between them in virtue of which Glasgow is west of Edinburgh? The relation can’t be in one city at the expense of the other, nor in each of them taken separately, since then we lose sight of the fact that the relation holds between them (McTaggart 1920: §80). Rather the relation must somehow share the divided locations of Glasgow and Edinburgh without itself being divided.

There may be a relation between 1 and 1 without there being a force between them.

This doesn't seem to accord with typical uses of 'cause'. Without oxygen in my room, I couldn't write this reply. But oxygen did not cause me to write it.

Without 1, 2 could not exist, though the reverse doesn’t hold. Since it is because of the existence of 1, or one thing, that there can be 2, or two things, then the former can be said to be the cause of the latter. — Pretty

Without meringue, the Australian dessert containing meringue, whipped cream and fruit couldn't exist.
The Australian dessert containing meringue, whipped cream and fruit is named Pavlova.
Therefore, without meringue, the Pavlova couldn't exist.
The Pavlova couldn't exist without meringue, because by definition, a Pavlova contains meringue.

Without 1, 1 + 1 couldn't exist
1 + 1 is named 2
Therefore, without 1, 2 couldn't exist
2 couldn't exist without 1, because by definition, 2 is 1 + 1

As "Pavlova" is a name, "2" is a name.

As meringue didn't cause the Pavlova, 1 didn't cause 2.

"Sherlock Holmes" is also a name. That something has a name doesn't of necessity mean that it exists in the world. We can talk about "Sherlock Holmes" even though "Sherlock Holmes" doesn't exist in the world. We can talk about 2 even though there is no necessity that 2 exists in the world.

That we can talk about 2 does not necessarily mean that 2 exists in the world.

Not really. They're mental in the way of being an interpretation of reality, but the categorization of things still end up in amounts. We can argue about how categories are human constructs, but at some point we get to things like 1 atom, 2 atoms. In relation to what numbers represent you cannot have 2 atoms if you didn't have 1 atom first. The same kind of works the other way around, how can you define something as 1 object if there wasn't the possibility of there being 2? — Christoffer

Counting doesn't have to start from 1 always. It can start from 2n, where n = 1/2. As Pantagruel suggested, if we suppose counting is a process, you don't even need things such as particles. They would be just the elements in the counting process, or sets.

Likewise, you don't always count each individual object which is 1. When you say 1, it might be 2 in reality. In the case of your shoes, or soaks, you count them as 1 pair of shoes, but there are 2 shoes in the pair.

. If you have 2, you have 9, and 5 and 4 and 1. — Christoffer

Could your explain what you mean by this?

The interesting thing, however, is whether or not "0" has a relation. That concept has more of a constructed meaning than single existence. What is "0.5"? Is it half of a one thing, or is it half of nothingness? — Christoffer

0 is just a description of objects or states of nothingness. It is a very handy concept in math.

Ok. How about this. Numbers primitively seem to correlate with things. But are there in fact things? Or are there really only processes, whose synchronic slices appear intermittently as things? In which case, numbers would really correlate with processes. Or again, we can only count insofar as we abstractly identify the things being counted. So we count one-hundred peanuts. Be we don't count one-hundred "things" as one-peanut, two-jar, three-house, four-planet, five-universe....etc. Numeracy is itself just the culmination of abstraction. Short of objective correlation, what inherent reality do numbers have except the cumulative set of interrelations which are defined by all the possible mathematical constructs in which they appear? — Pantagruel

Counting can be a process. So, yes, we can see the things being counted as the elements of the process. But you know, you can still count without things. What does it tell you? Numbers are not the things themselves. Numbers are real of course, but they are real in the sense that we know them, use them and apply them to the external world objects, events and motions, as well as we can think about them, and demonstrate them as pure concepts.

Numbers are not just the culmination of abstractions. Numbers can describe the whole universe as long as you know how they work. You can make up formulas, equations and axioms and replace the variable with the numeric data, from which you can understand the workings of the universe.

Ok this explanation as made the most sense to me. — Pretty

:up: :cool:

As a cause, it necessary implies the existence of its effect, yes? So let’s take a person who is a parent — surely as a person they exist far before their child, and their child does not have to necessarily exist, but as a *parent*, a causal thing, it is necessarily implied that their effect exists too, which we call the “child.” Is this correct? — Pretty

Cause and effect theory is a scientific concept. If you say A caused B, then whenever there was A, then B must follow in all occasions. Here the important point is that A must produce the exact same state, entity or result or effect condition B on all occasions.

Therefore your example of a parent X producing the child X1, is not a cause and effect relationship. Because the parent cannot willfully produce the child X1 next time they try to produce X1. No persons in the universe are exactly the same in the universe, and every person is unique in their identity by the law.

The X might produce another child X2, or may not produce any child at all, or might produce twins next time called X2 and X2a.

Therefore the offspring X1 is not an effect of the parent X under the eyes of causal relationship. They are a parent and offspring relationship.

Numbers are not just the culmination of abstractions. — Corvus

Here is an excerpt from R.G. Collingwood's Speculum Mentis on the logical nature of mathematical concepts, which emerge through the power of abstraction from experience (kind of Kantian I guess):

...the only really a priori or pure concept is the concept of a class as such, the concept of classification or abstraction....each member being simply another instance of the universal. This indeterminate plurality of units is precisely the numerical series. Each unit is distinguished from the rest simply as being another that is, by its ordinal number, and the common nature of units in general is simply that they are that of which there is an indeterminate multiplicity. This indeterminate multiplicity is the mathematical infinite, which is therefore another name for the perfect abstractness of the mathematical universal...a mere plurality of abstract units...Mathematics implies the ideal reduction of what are really unique facts to mere units.

:ok: :sparkle:

each member being simply another instance of the universal..............This indeterminate multiplicity is the mathematical infinite (RG Collingwood). — Pantagruel

There are an infinite number of possible numbers, such as 1, 1.1, 1.11, 1.111, etc.

If numbers exist in the world, they must exist either as abstract entities, such as 1, 2, 3, etc or concrete entities, such as 1 atom, 2 atoms, 3 atoms, etc.

Suppose 2 exists as a concrete entity, such as 2 atoms. As there an infinite number of possible numbers, but only a finite number of concrete entities in a finite world, then there are some possible numbers that cannot exist in the world. In this event, a mathematical infinite in the world is not possible.

A mathematical infinite can only exist in the world if numbers exist as abstract entities, independent of any concrete entities. This raises the question as to what relates the number 2 to 2 atoms rather than relating the number 2 to 5 atoms, for example?

Indeed. Obviously there is not a unique set of two "proto-digmatic" entities. On the other hand, any pair of things can exist in a state of "two-ness" given the appropriate abstraction. Which is Collingwood's rationale, I think. His metaphysics consists of a state of mutual inter-expression, where the individual exists in and through the universal, and vice-versa.

Cause and effect theory is a scientific concept. If you say A caused B, then whenever there was A, then B must follow in all occasions. Here the important point is that A must produce the exact same state, entity or result or effect condition B on all occasions. — Corvus

Cool, I’ve been slowly gathering this as the thread continued, I’m surprised it took this long to get explicated. Thanks!! It really does clear up a lot

Obviously there is not a unique set of two "proto-digmatic" entities.........................On the other hand, any pair of things can exist in a state of "two-ness" given the appropriate abstraction. — Pantagruel

I don't know what a "proto-digmatic" entity is.

Does two-ness exist in the world or in the mind of the observer?

Suppose two-ness exists in the world.

If two-ness exists in the world, then so must one-ness.

Suppose an observer sees two things in the world that are spatially separate.

What determines whether there is one two-ness or two one-nesses?

IE, if two-ness exists in the world, how does a particular thing in the world "know" whether it is related to another thing or not?

Prototypical. Paradigmatic. Proto-digmatic. Just having fun with language.

I think the essence of the answer regarding the nature of abstraction and the mutual inherence of the universal and the particular already addresses your questions. (i.e. twoness is simultaneously abstract but qua concrete instantiation). It sounds as if you basically don't agree with the characterizations of the particular and the universal-abstract that I'm embracing. The long quote I made from Collingwood is its own best evidence and equates with my claims.

The long quote I made from Collingwood is its own best evidence and equates with my claims. — Pantagruel

Collingwood also says:

Mathematics is thus the one and only a priori science. It has nothing to do with space or time or quantity, which are elements of concrete experience ; it is simply the theory of order, where order means classificatory order, structure in its most abstract possible form.

This seems to suggest that for Collingwood, numbers, being part of mathematics, exist in thought rather than sensation.

This seems to suggest that for Collingwood, numbers, being part of mathematics, exist in thought rather than sensation. — RussellA

True. Except that he relentlessly fuses these:

The concept is not something outside the world of sensuous appearance it is the very structure or order of the world itself....The universal is only real as exemplified in the particular, the particular as informed by the universal.

Which really is the case. We never experience vacant materiality, or pure conceptuality. However we do have abilities that seem to operate on a spectrum of synthesis that lies between these poles.

Cool, I’ve been slowly gathering this as the thread continued, I’m surprised it took this long to get explicated. Thanks!! It really does clear up a lot — Pretty

That sounds pretty cool, Pretty. Thanks for the great OP. :up: :cool:

I doubt it's simple as that in this world. — Gmak

Certainly not on this forum. :grin:

True. Except that he relentlessly fuses these: — Pantagruel

The fusing of thought and sensation. A seemingly Kantian approach, where the principles of pure understanding allow the very possibility of experience (CPR B293).

Collingwood writes in Speculum Mentis

Again, when I speak of a sensation, imagination, thought, or the like, I sometimes mean an
object sensated, sometimes the act, habit or faculty of sensating it, and so on.

Such thought and sensation exist in the mind, rather than outside the mind as things-in-themselves.

Collingwood writes "Mathematics is thus the one and only a priori science", inferring that, for Collingwood, numbers, as part of mathematics, exist in the mind rather than outside the mind.

Yes, it's useful to distinguish between them. Causes would involve individual instances, principles every case of twoness, a binary, etc.

Here is a quick explanation I wrote a while back:

The epistemic issues raised by multiplicity and ceaseless change are addressed by Aristotle’s distinction between principles and causes. Aristotle presents this distinction early in the Physics through a criticism of Anaxagoras. Anaxagoras posits an infinite number of principles at work in the world. Were Anaxagoras correct, discursive knowledge would be impossible. For instance, if we wanted to know “how bows work,” we would have to come to know each individual instance of a bow shooting an arrow, since there would be no unifying principle through which all bows work. We cannot come to know an infinite multitude in a finite time (for the same reason that one cannot cross an infinite space in a finite time at a finite speed.)

However, an infinite (or practically infinite) number of causes does not preclude meaningful knowledge if we allow that many causes might be known through a single principle (a One), which manifests at many times and in many places (the Many). Further, such principles do seem to be knowable. For instance, the principle of lift allows us to explain many instances of flight, both as respects animals and flying machines. Moreover, a single unifying principle might be relevant to many distinct sciences, just as the principle of lift informs both our understanding of flying organisms (biology) and flying machines (engineering).

For Aristotle, what are “better known to us” are the concrete particulars experienced directly by the senses. By contrast, what are “better known in themselves” are the more general principles at work in the world. Since every effect is a sign of its causes, we can move from the unmanageable multiplicity of concrete particulars to a deeper understanding of the world.For instance, individual insects are what are best known to us. In most parts of the world, we can directly experience vast multitudes of them simply by stepping outside our homes. However, there are 200 million insects for each human on the planet, and perhaps 30 million insect species. If knowledge could only be acquired through the experience of particulars, it seems that we could only ever come to know an infinitesimally small amount of what there is to know about insects. However, the entomologist is able to understand much about insects because they understand the principles that are unequally realized in individual species and particular members of those species.

But there is no inside without outside. Collingwood's position falls directly within the parameters of a philosophy of embodiment. He is the metaphysical grandfather of the idea of the embodied mind.

He is the metaphysical grandfather of the idea of the embodied mind. — Pantagruel

Do you have a source for this?

That an organism is embodied in the world does not mean that the organism necessarily has knowledge about the world.

That an organism is embodied in the world does not mean that the organism necessarily has knowledge about the world. — RussellA

Actually that is exactly what embodied-embedded cognition implies, represents a definition of knowledge as much as anything.

The idea that he is the metaphysical grandfather of embodied cognition is my own. Informed by having read five of his books as well as two extensive critical studies.

Actually that is exactly what embodied-embedded cognition implies, represents a definition of knowledge as much as anything. — Pantagruel

Embodied cognition is knowledge of interactions with the environment, not knowledge about what in the environment caused those interactions

Embodied cognition is the idea that the body or the body’s interactions with the environment constitute or contribute to cognition (SEP - Embodied Cognition)

This is why embodied cognition has been inspired by the phenomenological tradition

Another source of inspiration for embodied cognition is the phenomenological tradition. (SEP - Embodied Cognition)

Literally, phenomenology is the study of “phenomena”: appearances of things, or things as they appear in our experience, or the ways we experience things, thus the meanings things have in our experience. (SEP - Phenomenology)

In Collingwood's terms, it is knowledge about the sensations, not whatever in the world caused those sensations.

In Kant's terms, it is knowledge about Appearances, not knowledge about Things-in-themselves.

In language, the clause "that Lydia sang" is embedded within the clause "Wanda said that Lydia sang". The embedded clause "that Lydia sang" gives no information about the clause it is embedded into, "Wanda said that Lydia sang"

In geology, silver may be embedded in copper. The embedded silver gives no information about the copper it is embedded into.

Embodied cognition has knowledge, but knowledge of thoughts and sensations, not knowledge about what in the world caused those thoughts and sensations.

Embodied cognition is the idea that the body or the body’s interactions with the environment constitute or contribute to cognition...

The sentence continues: in ways that require a new framework for its investigation. The first part has rarely been denied, although it is sometime more or less ignored.

The enactivists I am aware of tend to be harsh critics of Kantian representationalism. It gets offered up as a way to avoid Kant's problems, not a way to recreate them. The article you're citing mentions phenomenology as a means of dissolving the very Kantian dualism you are claiming this approach represents.

Embodied cognition is knowledge of interactions with the environment, not knowledge about what in the environment caused those interactions — RussellA

This is a misconstrual of embodied cognition, which is not about "knowing that" at all. It's about knowledge being enacted via its environmental embeddings, and extends outward, rather than inward, as in the associated concept of distributed cognition, where environmental features are construed as being actual elements of cognitive processes.

However this isn't the place to address that as we are veering OT for this thread.