No, numbers do not have causal efficacy. They are not efficient causes, in any sense of the term.

I can get why they’re not efficient causes at least, but I’m trying to grasp this in the same lens that the Aristotelian tradition considered the genus of a thing to be the cause of its species. Now, 1 is obviously no genus of 2, but is the genus in any way argued as *efficient* cause by them, or is it formal?

And regardless of that, is it at least then established and agreed upon by most experts that a thing can be necessary for the existence for another thing, and yet not be a cause? If so, my more confused question would be what best defines a cause most generally across all types besides this criteria of necessary priority?

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

Presumably "exist" is referring to existing in the world rather than existing in the mind.

2 is the relation between 1 and 1.

The question to ask is, do relations ontologically exist in the world.

If relations don't ontologically exist in the world, then there is no relation between 1 and 1. This means that there is no 2. As there is no 2, the concept of did 1 and 1 cause 2 is not applicable.

If relations do ontologically exist in the world, then there is a relation between 1 and 1. However, such a relation is contemporaneous with 1 and 1. On the one hand, the relation between 1 and 1 didn't exist prior to there being a 1 and 1 (as the relation is between 1 and 1) and on the other hand, the relation between 1 and 1 didn't exist subsequently to there being a 1 and 1 (otherwise for a moment in time there would have been no relation between 1 and 1). As the relation between 1 and 1 is contemporaneous with 1 and 1, the concept of cause is not applicable.

Either way, whether relations do or do not ontologically exist in the world, the concept of cause is not applicable to numbers.

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

Does this hold? Surely this argument has been made plenty times before, no? — Pretty

No. it doesn't. Number can start from any number you decided to choose to start. Because numbers are the mental concept. There is no physical laws or principles on numbers.

After 1, counting can go on via the real or rational numbers never reaching 2. Or counting can proceed in the odd numbers skipping 2, and all the even numbers.

If 1 caused 2, then every time 1 appears immediately 2 must appear, if they have cause and effect relationship. But it doesn't. You order 1 coffee in the caffee, and you don't see 2 coffees served to you unless by mistake or confusion of the maid.

1 is a property of an object saying it only stands as 1. 2 only appears when there are 2 objects, and counted.

No, numbers do not have causal efficacy. They are not efficient causes, in any sense of the term. — Arcane Sandwich

What about considering binary fission as exemplifying a kind of organic ontology. One parent cell is the efficient cause of two daughter cells. One is the cause, two are the effects.

No. it doesn't. Number can start from any number you decided to choose to start. Because numbers are the mental concept. There is no physical laws or principles on numbers. — Corvus

Numbers can be mental concepts. However anything natural can also exist as a mental concept. And numbers appear to inhere in the natural world, as evidenced by the existence of mathematizable relationships. So what basis is there for claiming numbers are, or numeracy is, exclusively mental?

can also exist as a mental concept. — Pantagruel

Maybe. I just don't see the point saying mental objects "exist". It is a misuse of language.
We know, or are aware of the mental objects. They don't exist like the physical objects in the external world.

I am trying to see the existence of "3" in the external world. I see none. I can see 3 books, 3 cups, 3 trees, 3 cars. But none of them are the pure "3".

In China, 3 is not the real 3 either. The real 3 is written as "三".
In Korea, 3 is written "셋".
Now, which is the real 3?

We know, or are aware of the mental objects. They don't exist like the physical objects in the external world. — Corvus

Even if that were true, it wouldn't contradict the existence of an objective correlate of the mental object. i.e. Just because numbers have a mental appearance, doesn't mean that numeracy isn't a physical reality. My go-to example is the use of Fibonacci-sequence timed laser pulses to stimulate atoms into a new phase state of matter. Nature is "resonant" to numerical properties....

My go-to example is the use of Fibonacci-sequence timed laser pulses to stimulate atoms into a new phase state of matter. Nature is "resonant" to numerical properties.... — Pantagruel

Sure, numbers describe the external objects, events and motions. But it is an illusion to think they are the same, or numbers are the physical reality. Math formulas, equations and functions are descriptions of the physical world. Description is not physical objects.

For example, the word apple is not the real apple. You cannot eat the word apple. You can only eat the real apple which can be peeled. To say the word apple is same as the real apple is an illusion.

Physical reality is the things and objects you can see, touch, access and physically handle.

Math formulas, equations and functions are descriptions of the physical world. Description is not physical objects. — Corvus

The sun is yellow. Yellow is not a physical object. But the light being emitted at 510 Terahertz is.

The sun is yellow. Yellow is not a physical object. But the light being emitted at 510 Terahertz is. — Pantagruel

If you wore a green sunglasses, and look up at the sun, it will look "green". When I measure the light of the sun with the optical light meter, it says f16 1/1000 sec.

Exactly. Mathematical relationships inhere in material objects. The abundance of fractal features in the universe additionally is suggestive of this possibility. It's just an empirical observation for me. But I see no reason to discount the reality of numbers. Ipseity may be the foundation of all logic.

It's just an empirical observation for me. But I see no reason to discount the reality of numbers. — Pantagruel

Sure. Some folks believe God is the absolute reality, or the big bang is the absolute truth. One's belief can be real to the believer, but it can be irrational and illogical too.

I am not saying numbers are false, or unreal. All I am saying is it has different mode of reality i.e. we know numbers as concepts and use them to describe the real world objects, motions or workings. But they don't exist like the physical objects do.

But they don't exist like the physical objects do. — Corvus

You mean like quantum fields, that kind of "substantively real" thing? Or more like statistically defined entities like subatomic particles?

I think numbers are like words i.e. adjectives describing the objects such as red apple. The red is an adjective describing the apple's colour. When we say 1 apple. 1 describes the apple's existence i.e. the quantity which is 1.

If you look at the Hebrew language, they don't have number system. Words are also numbers.
Numbers are descriptive language of the objects and motions in quantity in existence. It is purely psychological and conceptual descriptive tool.

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?

As the relation between 1 and 1 is contemporaneous with 1 and 1, the concept of cause is not applicable.

Hold on, this doesn’t feel unanimously agreed upon. Aristotle speaks of a certain priority in which two things exist contemporaneous to each other yet still have a causal-effective relationship — such as the existence of a thing and an affirmation of that thing. Similarly, when Spinoza gives a causal nature to substance as substance, he does not imply that substance ever existed before its modes at large. While one mode may come and go, “modes” as a whole are inextricable from substance. And yet, for Spinoza, substance is “prior in nature” to modes and causal of them.

Are you saying it can’t be argued through their thought that mutually contemporaneous things are causal of each other?

If 1 caused 2, then every time 1 appears immediately 2 must appear, if they have cause and effect relationship. But it doesn't. You order 1 coffee in the caffee, and you don't see 2 coffees served to you unless by mistake or confusion of the maid. — Corvus

Ok this explanation as made the most sense to me. Idk if get down with the non-reality of numbers but this part is what matters to me. 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?

So in a strict sense, would we say that a thing, in order to properly understand it as a cause, *must* have an existent effect that followed necessarily from some specific aspect of that thing, and it is precisely this specific aspect which, through its necessary bringing about of the effect, we would call the cause? I can get down with this but doesn’t it kind of restrict causality to the realm of determinism? Is there a way we can understand, for example, one person being the cause of another’s actions, but where those actions were in no way necessary to follow and largely came about from the will of that other person? Or is this just simply not a cause in our strict sense?

Because numbers are the mental concept. — Corvus

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? You cannot form the interpretation of reality into an object existing as the only 1 object in existence if there wasn't a relation to more than one. So naturally, math has a backwards causation in that math, as an interpretive system requires the whole system in order to form a "1".

On the other hand, that may constitute that there's no causality for the existence of numbers in order, but rather that if you have 1, you also have all other numbers when using math in our reality. If you have 2, you have 9, and 5 and 4 and 1.

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?

The concept of non-existence is therefore much harder to correlate with a causal connection in math. Maybe that's why math using infinity end up so confusing for everyone. We fundamentally operate in a reality where everything exists and there's no physical representation of absolute nothingness.

Perhaps a more interesting version of this question is to ask, "Does the addition of 2 and 2 cause the result 4?" That is certainly not how we speak about it ordinarily, probably because we limit our concept of causation to the spatio-temporal world. So what about this version?: "Does my thought of the addition of 2 and 2 cause me to conclude that the result is 4?" (if you're willing to accept "my thought" as an event in space and time, not as a Fregean proposition).

Sure, this feels like it’s clearing things up. So taking the assumption I’m making from the other commenter, if a cause necessarily leads to its effect, it makes sense how two and two necessarily lead to four, while two by itself does not necessarily lead to it at all. So the bringing together of 1 and 1 and 1 and 1 is the cause of 4, but 1, 2, 3, or any other smaller number by themselves can’t cause 4. Similarly, while a good parent has the possibility of bringing about a fair child, the good parent on their own cannot produce the child without another parent, and the specific composition of the one parent determines the necessary composition the second parent must have in order for the child to become fair? And so properly spoken, the fair child’s cause is not the one good parent, but the bringing together of that one parent with another appropriate one. This bringing together necessarily leads to a child who has a fair disposition, and so as a whole process is the cause, but not the process’ components. Does this make sense or am I talking nonsense here?

if a cause necessarily leads to its effect, it makes sense how two and two necessarily lead to four, while two by itself does not necessarily lead to it at all. So the bringing together of 1 and 1 and 1 and 1 is the cause of 4, but 1, 2, 3, or any other smaller number by themselves can’t cause 4 — Pretty

That would be one answer to what I was calling a more interesting question, yes. Addition does seem a more plausible candidate for causal efficacy than mere sequence. But does any of this really work? You use the term "lead to" to describe what a cause does, re its effect, but I think we have to make it stronger, and say forthrightly that a cause causes the effect, it doesn't just "lead to it" in some weaker way. On that understanding, I don't see numbers, even when added, multiplied, etc., causing their results. This may just be a spade-turning commitment on my part to viewing cause as separated in time from effect.

But that was why I then moved on to thoughts as causes. In a functionalist, psychological way, we can talk about thought A (viewed as a brain-event) causing thought B, even though as yet our science doesn't really know what this means. The question is, is that the same kind of "causing" that we mean when we say that "my thought of A" causes B? We want to say that thought A justifies or explains, rather than causes, thought B -- but that is to bring in the Fregean notion of a thought/proposition that can be abstracted from any given instance of its occurrence in a brain.

This may just be a spade-turning commitment on my part to viewing cause as separated in time from effect. — J

To me, causality does seem prior to time itself, as time seems entirely reliant on more basic and general ideas of order and necessity, but I understand the effort :)

But that was why I then moved on to thoughts as causes. In a functionalist, psychological way, we can talk about thought A (viewed as a brain-event) causing thought B, even though as yet our science doesn't really know what this means. The question is, is that the same kind of "causing" that we mean when we say that "my thought of A" causes B? We want to say that thought A justifies or explains, rather than causes, thought B -- but that is to bring in the Fregean notion of a thought/proposition that can be abstracted from any given instance of its occurrence in a brain. — J

Well it seems to me we can do the same logic I employed in the last comment, no? Wouldn’t “Thought A” simply be part of a fuller composition of reality, which, when considered altogether as a whole, gives an account for why “Thought B” necessarily followed? Surely Thought A on its own can be shown to not lead to Thought B in plenty of other contexts, but in the specific context in which it *does* come about, wouldn’t Thought A then be both necessary to the existence of Thought B, as well as albeit only a piece of the fuller composition that led to this existence of Thought B? Insofar as B, as an effect, can be logically understood through *some* cause, wouldn’t we be safe to think there is some formal general consistency to the certain compositions of reality that bring about Thought B, so that in this sense Thought A is simply filling a role that other thoughts in the past have filled in also causing Thought B?

For example, a thought about a loved one’s deceit (Thought A) might make me have a thought of hopelessness (Thought B), and this thought is likely necessarily furnished by other thoughts, perhaps about the history of my life (Thought C) in order for Thought B to be caused. But many other collections of thoughts could have led me to the same Thought B, such as a thought of getting fired (Thought F) and a thought of a dismal future (Thought G). So A+C causes B, but F+G also causes B. So in this case it would be more appropriate to understand the logical consistency between A+C and F+G as respective pairs, to see what aspects of their individual collections exist essentially as the same cause of the same effect. This consistent aspect, that we would assumably find in all other thought-complexes that lead to Thought B, could be considered the formal cause of thought B, so we may cause it Forms X+Y, where Form X is the role filled by Thoughts A and F respectively, while Form Y is the role filled by thoughts C and G.

So to speak properly, the proper cause of Thought B into reality is the bringing together of Form X and Form Y through their instantiations, whether it be Thoughts A+C, or Thoughts F+G, and this is precisely the role that Thought A has in the cause of Thought B. Further, we can simplify this language by saying Forms X+Y together make up Form Z, which is simply the analogous Form of Thought B in all of its instantiations, and what we utilize when we understand two completely distinct thoughts as both for some reason qualifying “Thought B”

Aristotle speaks of a certain priority in which two things exist contemporaneous to each other yet still have a causal-effective relationship — such as the existence of a thing and an affirmation of that thing. — Pretty

As regards Aristotle's Material Cause, which is an intrinsic cause, for example a table is made of wood and a statue is made of bronze.

I agree that the table is contemporaneous with the wood it is made from, and is described by Aristotle as a cause.

Aristotle also describes an Efficient Cause, which is an extrinsic cause, for example a sculptor who chisels at a block of marble to transform it into a statue.

The OP asks "Is the number 1 a cause of the number 2?"

There are different meanings to the word "cause", whether intrinsic cause or extrinsic cause.

If ontological relations don't exist in the world, then 2 cannot exist, meaning that there can be no cause of 2 whether intrinsic or extrinsic.

If relations do exist, taking the example of Material Cause, as a table is made of wood, 2 is made of the relation between 1 and 1.

The next question is, are there any good reasons for supposing that ontological relations do exist in the world?

The next question is, are there any good reasons for supposing that ontological relations do exist in the world? — RussellA

Wouldn’t gravity be a perfect example of one?

What about considering binary fission as exemplifying a kind of organic ontology. One parent cell is the efficient cause of two daughter cells. One is the cause, two are the effects. — Pantagruel

Then you run into the paradox of the Ship of Theseus, is what I would say. Which is a problem of indeterminate identity.

↪Arcane Sandwich
I can get why they’re not efficient causes at least, but I’m trying to grasp this in the same lens that the Aristotelian tradition considered the genus of a thing to be the cause of its species. Now, 1 is obviously no genus of 2, but is the genus in any way argued as *efficient* cause by them, or is it formal?

And regardless of that, is it at least then established and agreed upon by most experts that a thing can be necessary for the existence for another thing, and yet not be a cause? If so, my more confused question would be what best defines a cause most generally across all types besides this criteria of necessary priority? — Pretty

These are very difficult questions that you're asking, and I don't have an opinion on such matters at the moment. I am, however, actively working on those topics, on paper. That's all I can say.

fair enough, thanks for your contributions :)

↪Arcane Sandwich I can get why they’re not efficient causes at least, but I’m trying to grasp this in the same lens that the Aristotelian tradition considered the genus of a thing to be the cause of its species. Now, 1 is obviously no genus of 2, but is the genus in any way argued as *efficient* cause by them, or is it formal?

And regardless of that, is it at least then established and agreed upon by most experts that a thing can be necessary for the existence for another thing, and yet not be a cause? If so, my more confused question would be what best defines a cause most generally across all types besides this criteria of necessary priority?

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? That might be the place to start, but that's covered more in the Physics (Joe Sachs guided translation has some good stuff on this). Book V on causes is relevant too.

https://plato.stanford.edu/entries/aristotle-mathematics/#10 <== is also pretty relevant.

Key points:

Numbers are more like concatenations of units and are not sets. To draw a contrast with modern treatments of numbers, a Greek pair or a two is neither a subset of a triple, nor a member of a triple. It is a part of three. If I say that ten cows are hungry, then I am not saying that a set is hungry. Or to point to another use of ‘set’, my 12 piece teaset is in a cabinet, not in an abstract universe. So too, these ten units are a part of these twenty units:

One (a unit) typically is not a number (but Aristotle is ambivalent on this), since a number is a plurality of units.

See also: Metaphysics 1052b35: http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0052%3Abook%3D10%3Asection%3D1052b

Unit is related to number as principle, not species.