Is There a Cure for Metaphysical Dogma?

[Nous]

it's called science.

So what were these claims about: "Science isn't something to believe in. It's theories aren't intended to impress truth upon the world."?

Were you just trying to knock down some straw men?

I take it that you cannot articulate any argument contrary to the Indispensability Arguments that deduce mathematical realism from scientific realism.

 

[Nous]

the relata in my example are the sizes of the particles, not the particles themselves, and the ratio of the sizes is the relation.

So you do not believe that particles have any "ontological reality". So much for physicalism.

What physicists discover about empirical reality are relations between quantities. These are mathematical relations such as equals ( = ) or greater than or less than ( >, < ). The relation between the respective "size" of 2 particles is obviously a mathematical relation.

So you have no arguments contrary to the thesis of mathematical realism as described in the quoted paragraph of the Routledge article? And you cannot show any error in the Quine-Putnam Indispensability Argument or any of my versions of it in the OP here: Scientific Realism Begets Mathematical Realism, in which mathematical realism is implied by scientific realism?

 

[Willamena]

So what were these claims about: "Science isn't something to believe in. It's theories aren't intended to impress truth upon the world."?

The epistemological method at our disposal to discover reality is not something to believe in. To believe means to invest genuinely in its truth. We have another word for people who do that in regards to science: scientism.

Its theories aren't intended to impress truth upon the world. They are intended to be flexible, to grow and evolve with new learning. That they are epistemic in nature points to their falsifiability, they are entirely dependent upon the level of knowledge used in their construction.

 

[LukeS]

You got me. I'm not familiar with 1-valued or zero-valued logic. What are those things?

Well if metaphysics is a disorder, maybe its all the same, of zero logical value???

I am not sure metaphysics needs to be cured.

 

[siti]

So you have no arguments contrary to the thesis of mathematical realism as described in the quoted paragraph of the Routledge article? And you cannot show any error in the Quine-Putnam Indispensability Argument or any of my versions of it in the OP here: Scientific Realism Begets Mathematical Realism, in which mathematical realism is implied by scientific realism?

I'm sorry - how is pointing out that the argument is based on a weak first premise and a second premise that begs the question - thereby making the argument circular and the conclusion, at best, a best guess based on the assumption that mathematical statements are real "entities" without presenting any argument or evidence to establish this... not a counter-argument?

I have already explained how the indispensability argument fails and you are not responding to the content of my post - you are just restating your unfounded contention that I have no counter-argument even when I have clearly presented one. So you are either still very confused - in which case I am wasting my time, or you are deliberately ignoring the clear evidence before you and sticking to your flimsy argument - in which case I am wasting my time.

So far in this thread you have merely demonstrated the very kind of dogmatic assertions that are symptomatic of the "metaphysical dogma" you identified as the problem and railed against at the beginning. I have listened patiently to the "kettle calling the pot black" and watched as you unwrapped your dogmatic devotion to the metaphysical thesis of "mathematical realism" for all to see. Now - I am going to ask you the questions you put to us in your OP:

On what grounds (other than the flimsy and circular reasoning of the so-called "indispensability argument") have you concluded its truth?

Is this thesis falsifiable?

If so, what fact or evidence would falsify it?

If you can't present a convincing argument that is not premised on its own conclusion, then there is no question that your preference for mathematical realism is every bit as dogmatic as any other and I'm not going to waste any more time on this topic.

 

[Nous]

The epistemological method at our disposal to discover reality is not something to believe in. To believe means to invest genuinely in its truth. We have another word for people who do that in regards to science: scientism.

Its theories aren't intended to impress truth upon the world. They are intended to be flexible, to grow and evolve with new learning. That they are epistemic in nature points to their falsifiability, they are entirely dependent upon the level of knowledge used in their construction.

So, your claims--"Science isn't something to believe in. It's theories aren't intended to impress truth upon the world"--were no in response to anything anyone has said on this thread? You just wanted to knock down a couple of straw men?

I asked you whether you believe there is any epistemological method available to humans by which to discover reality, to which you answered "it's called science." So do you believe that "science" is an epistemological method by which to discover reality, or not?

 

[Nous]

Well if metaphysics is a disorderof zero logical value???

I am not sure metaphysics needs to be cured.

Who says "metaphysics needs to be cured"? It's metaphysical dogma that's irrational--as several people on this thread have demonstrated.

 

[Nous]

I'm sorry - how is pointing out that the argument is based on a weak first premise and a second premise that begs the question - thereby making the argument circular

What argument do you claim is "circular"? I am unaware that a valid syllogism can be circular.

I have already explained how the indispensability argument fails

I haven't seen any such explanation. Are you saying that all 3 syllogisms in the OP here "fail": Scientific Realism Begets Mathematical Realism ?

Are you claiming that these arguments are unsound or invalid?

you are just restating your unfounded contention that I have no counter-argument even when I have clearly presented one.

Quote the argument you have presented. I didn't see it.

P1: [. . . ]
P2: [. . . ]
C: Therefore, the thesis of mathematical realism is false.

Is this thesis falsifiable?

If so, what fact or evidence would falsify it?

Of course, the indispensability arguments in the thread above are falsifiable. All one needs to do is to show that either the minor or major premise in any of the arguments is false, or else show that the sylllogism is invalid.

 

[Willamena]

So, your claims--"Science isn't something to believe in. It's theories aren't intended to impress truth upon the world"--were no in response to anything anyone has said on this thread? You just wanted to knock down a couple of straw men?

If you like.

I asked you whether you believe there is any epistemological method available to humans by which to discover reality, to which you answered "it's called science." So do you believe that "science" is an epistemological method by which to discover reality, or not?

That is what I said, yes.

 

[Nous]

If you like.

You are welcomed to reveal to us your purpose for making assertions that are not responses to anything anyone has said on this thread.

 

[siti]

Of course, the indispensability arguments in the thread above are falsifiable. All one needs to do is to show that either the minor or major premise in any of the arguments is false, or else show that the sylllogism is invalid.

Both of which I have already done in previous posts - but I'll state it again here:

1. an entity is usually defined as a thing with independent or distinct existence

2. for mathematical statements to be "entities" they must therefore have independent or distinct existence in their own right - this is tantamount to saying they are "ontologically real"...

...therefore, the conclusion of the indispensability argument is faulty because the conclusion is assumed in the premises by defining mathematical statements as "entities"

Try it - take away the word "entity" and replace it with "statement" - does the argument still hold? Of course not, because if it did then we could equally argue for the ontological reality of any attributive statement - such as a "greater than" or "smaller than" relationship. Do, say, "taller" or "fatter" have ontological reality?

Show me a mathematical "entity" - a bit of math that has its own independent existence.

Of course I cannot produce an argument that proves that mathematical realism is actually false - it may be true that some mathematical entities are ontologically real - or it may not be. The thesis is not falsifiable - it is a matter of belief...but the indispensability argument is unsound.

On the other hand, we could argue that mathematical statements are always about something. It really doesn't matter whether the "things" they are about are real entities or imaginary concepts but if they happen to be about real things then the mathematical statements are attributes of real entities but not real entities in and of themselves.

 

[Nous]

Of course, the indispensability arguments in the thread above are falsifiable. All one needs to do is to show that either the minor or major premise in any of the arguments is false, or else show that the sylllogism is invalid.

Both of which I have already done in previous posts - but I'll state it again here:

These are the two arguments that I stated in the OP of the thread Scientific Realism Begets Mathematical Realism :

(1)

P1: All central terms of fundamental scientific laws are genuinely referential.
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are genuinely referential.​

And I noted that it has the form AAI-3--which is:

All M are P.
All M are S.
Therefore, some S are P.

(2)

P1: All entities (/structures) discovered by physicists using the scientific method are objectively existing.
P2: Some mathematical relations are entities (/structures) discovered by physicists using the scientific method.
C: Therefore, some mathematical relations are objectively existing.​

And I noted that it's form is AII-1--which states:

All M are P.
Some S are M.
Therefore, some S are P.

And Colyvan rendered the Quine-Putnam indispensability argument as:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities​

Which can be made an AII-1.

But you claim that these arguments are not valid? Cite your sources showing that AAI-3 and AII-1 are not valid syllogisms.



What is this:

1. an entity is usually defined as a thing with independent or distinct existence

2. for mathematical statements to be "entities" they must therefore have independent or distinct existence in their own right - this is tantamount to saying they are "ontologically real"...

...therefore, the conclusion of the indispensability argument is faulty because the conclusion is assumed in the premises by defining mathematical statements as "entities"

?

Is that supposed to be a valid argument? If so, what form is it? Identify your subject, predicate and middle terms.

In none of the three arguments just stated is the conclusion contained in any premise. Right?

I am confident that as soon as you become knowledgeable of how to make a deduction, you will agree with my earlier comment that a valid syllogism cannot be "circular".

And I must say it does occur to me now that the reason you were never able to state an argument that concludes that the theses of either "naturalism" or "physicalism" are true is apparently because you don't know how to make a deduction.

 

[Willamena]

You are welcomed to reveal to us your purpose for making assertions that are not responses to anything anyone has said on this thread.

Okay.

 

[siti]

Jesus H...I was not attempting to write out a formal syllogism - that is not the only way to make an argument - let alone a counter argument. I was attacking a premise.

If you want to validate Colyvan's syllogism you have to define "mathematical entity" in a way that does not pre-suppose its ontological reality.

As for your reformulation of his first premise - you must know that it is absolutely preposterous to suggest that:

"all scientific realists are ontologically committed to all and only the entities that are indispensable to our best scientific theories"

which (bold part) is what you originally posted before editing the claim to an unsubstantiated assertion that such a reformulation could be done.


Reformulated like this as an AII-1 the argument would go:

P1: all scientific realists are ontologically committed to all and only the entities that are indispensable to our best scientific theories
P2: some mathematical entities are indispensable etc...
C: all scientific realists are ontologically committed to some mathematical entities

It is a valid syllogism iff P1 is true, but it is patently obvious that some scientific realists are not ontologically committed to all 'indispensable' "entities" because some scientific realists might very well hold that mathematical entities, whilst indispensable to our understanding of the world, are nothing more than conceptual formalisms or fictions about the ontologically real world that are indispensable only if we wish to "understand" the world but not genuinely ontological parts of it. In fact, they could argue that (at least some) mathematical 'objects' are not 'entities' at all and can have no independent or objective existence apart from either the objective reality they describe or the minds that conceive of them. Neither of these positions is contrary to scientific realism so P1 is not a valid premise.

Even in Colyvan's original form, the argument relies on an unexplained "ought" - why "ought" we "to have ontological commitment to all and only the entities that are indispensable to our best scientific theories"?

And P2 is questionable depending on what is meant by "entity". As I have now pointed out several times, if we are calling mathematical statements "entities" we are already assuming they have a distinct and independent existence - how is that not an "ontological commitment"? But this is in the premise, not the conclusion. At best, the language choice in this formulation reifies a concept prematurely. We are calling the "picture of reality" that math presents as if it were the reality itself. At worst, it renders the argument circular (as I said) by making the very ontological commitment in the premise that the argument is designed to establish as its conclusion. If you still don't see that - and I will not be surprised if you don't - I don't see any point continuing the discussion on the indispensability argument.

Moving to your other formulations:

P1: All central terms of fundamental scientific laws are genuinely referential.
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are genuinely referential.

This, assuming I understand your use of the term "referential" correctly, merely proves that some mathematical quantities and relations point to real entities in the external world - rather than being purely linguistic or conceptual constructs of the mind - but that does not prove that the mathematical 'objects' are themselves ontologically real - they could still be (as Whitehead and Russell once hoped) pointing to a deeper logical reality on which the math is constructed, or as a physicalist might claim, attributive or purely conceptual statements about the ontological realities they describe. Again, it is a valid syllogism (as long as we are not making too strong a claim with "genuinely referential" - but it does nothing to prove the veracity of the thesis of mathematical realism.

And the second...

P1: All entities (/structures) discovered by physicists using the scientific method are objectively existing.
P2: Some mathematical relations are entities (/structures) discovered by physicists using the scientific method.
C: Therefore, some mathematical relations are objectively existing.

...makes the same invalid assumption in the second premise that Colyvan's version of the indispensability argument makes - namely that mathematical relations are 'entities' - but unless one can prove that these mathematical relations have an independent reality this is not a valid assumption - and even if it is, it presupposes the ontological reality you are attempting to establish by means of the argument.

In this argument, I am not sure that your first premise is not really a tautology - in terms of this argument, what would be the difference between an "entity discovered by physicists" and an "objectively existing" thing? P1 in this argument looks more like a definition than a premise. Maybe it is the definition I have been asking for, in which case you are saying that "entities" are "objectively existing things that are discovered by physicists using the scientific method". In that case, perhaps you do have a valid argument for the "objective existence" of some mathematical relations - but is there any reason that "objective existence" entails or necessitates "ontological reality"? How so?

Quite apart from invalid premises etc...

Suppose I wanted to borrow from your last argument and use the same form of syllogism to establish the truth of the thesis of physicalism - I could try this:

P1: All entities discovered by physicists using the scientific method are objectively existing.
P2: Some entities discovered by physicists using the scientific method are physical objects (which I define as 'things' existing as matter and/or energy).
C: Therefore, some objectively existing things are physical objects.

Again this might be a valid syllogism (but that still hinges on the definition of 'entity'), and it is - unlike the arguments you presented for mathematical realism - unquestionably true that some objectively existing things are physical objects. But it proves absolutely nothing about the thesis of physicalism (I'm sure you will not disagree with that).

 

[Willamena]

In any case, as I have now pointed out several times, if we are calling mathematical 'things' "entities" we are already assuming they have a distinct and independent existence - how is that not an "ontological commitment"? But this is in the premise, not the conclusion.

^This.

 

[Nous]

(1)

If you want to validate Colyvan's syllogism

(2)

Reformulated like this as an AII-1 the argument would go:

P1: all scientific realists are ontologically committed to all and only the entities that are indispensable to our best scientific theories
P2: some mathematical entities are indispensable etc...
C: all scientific realists are ontologically committed to some mathematical entities

(3)

invalid premises

(4)

It is a valid syllogism iff P1 is true

Good lord, Siti, you really don't have a clue as to what you're talking about. You are pitifully unknowledgeable in the most basic principles of logic.

(1) There is no such activity as "validating" a syllogism. One can show that an argument is valid by simply identifying its form or the rule of inference that is used.

(2) The "reformulated" thing you stated is not a syllogism of any sort. Obviously you cannot identify the middle, subject and predicate terms. It is not a valid argument of any sort.

(3) A syllogism is not made valid by a premise being true. A syllogism can be valid while the premises are untrue statements--i.e., while the deduction is unsound. And premises can be true statements while the syllogism is invalid.

(4) Arguments or syllogisms are valid. Premises are not valid. Premises are either true or false propositions.

you have to define "mathematical entity" in a way that does not pre-suppose its ontological reality.

Neither the validity or soundness of the Quine-Putnam indispensability argument rests on some hat trick in which the meaning of "mathematical entities" camoflauges your adjectival phrase "ontologically real". That phrase does not occur in the argument. The QP indispensability argument is a valid syllogism, and valid syllogisms are not "circular". Apparently you have not been exposed to many valid arguments before now.

it is patently obvious that some scientific realists are not ontologically committed to all 'indispensable' "entities"

Actually that's exactly why I deleted my reformulation of the QP argument, and I just didn't have time to redo without stating that absurd premise. Sometimes it takes me a little while to arrange the terms correctly to make a syllogism. You should try it.

This, assuming I understand your use of the term "referential" correctly, merely proves that some mathematical quantities and relations point to real entities in the external world

Exactly like some words "point to real entities in the external world"--e.g., "particle".

but that does not prove that the mathematical 'objects' are themselves ontologically real

That's a vacuous claim. You don't know what your phrase "ontologically real" means--you haven't defined what makes an entity or whatnot "ontologically real". Besides that, the phrase is redundant.

No one argues that the thesis of mathematical realism is true because mathematical entities have volume and location in spacetime. Mathematical entities don't have those characteristics. In any case, not one of the arguments for mathematical realism that I have noted, including those in the quoted Routledge article, hinge on mathematical entities conforming to your airy-fairy phrase "ontologically real". The whole of your "objections" to the valid arguments and indisputable facts (noted by Routledge) relating to mathematical realism is one big confused straw man, that entails the absurdity that valid syllogisms are "circular".

Suppose I wanted to borrow from your last argument and use the same form of syllogism to establish the truth of the thesis of physicalism - I could try this:

P1: All entities discovered by physicists using the scientific method are objectively existing.
P2: Some entities discovered by physicists using the scientific method are physical objects (which I define as 'things' existing as matter and/or energy).
C: Therefore, some objectively existing things are physical objects.

Again this might be a valid syllogism (but that still hinges on the definition of 'entity'), and it is - unlike the arguments you presented for mathematical realism - unquestionably true that some objectively existing things are physical objects. But it proves absolutely nothing about the thesis of physicalism (I'm sure you will not disagree with that).

That's correct. That syllogism does not argue for physicalism because the thesis of physicalism asserts that only "physical" things exist. Mathematical realism is not a thesis that claims that only mathematical things exist. As I made clear on the thread Scientific Realism Begets Mathematical Realism, there is definitely no problem with the proposal that physicists discover other sorts of phenomena than mathematical relations.

...makes the same invalid assumption in the second premise that Colyvan's version of the indispensability argument makes - namely that mathematical relations are 'entities' - but unless one can prove that these mathematical relations have an independent reality this is not a valid assumption

What nonsense. (1) Assumptions, unless they are arguments, are neither valid nor invalid. (2) As the Routledge article makes clear, the fact that mathematics is discovered, not invented, by humans is one of the primary premises by which to deduce the thesis of mathematical realism. One cannot invent the answer to the Riemann Hypothesis; the answer must be discovered. The solution to the problem has not been discovered yet.

I was not attempting to write out a formal syllogism

You should try to learn to do so. Your thinking is unclear--you seem to believe that if you believe something

 

[siti]

Good Lord Nous! You are questioning my ability to reason logically? Here's what you wrote originally (verbatim):

And Colyvan rendered the Quine-Putnam indispensability argument as:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities

Which is just an AII-1 by substituting "all scientific realists are ontologically committed" for "we ought to have ontological commitment".

But you claim that these arguments are not valid? Cite your sources showing that AAI-3 and AII-1 are not valid syllogisms.

Here's what I did (exactly what you said):

Reformulated like this as an AII-1 the argument would go:

P1: all scientific realists are ontologically committed to all and only the entities that are indispensable to our best scientific theories
P2: some mathematical entities are indispensable etc...
C: all scientific realists are ontologically committed to some mathematical entities

I then noted that this was not a valid argument because P1 is not true - of course I should have said it is not a sound argument - but anyway that's what I meant. And you comment:

(2) The "reformulated" thing you stated is not a syllogism of any sort. Obviously you cannot identify the middle, subject and predicate terms. It is not a valid argument of any sort.

It is a valid argument. The subject is "scientific realists", (or "we" in Colyvan's) the predicate is their (our) "ontological commitment" and the middle is the "indispensable entities" which is (as far as I can tell but I still don't know what is meant by "mathematical entities" but see below and previous posts) quite properly distributed (but see below). But it is not a sound argument because of the patently untrue first premise.

So now - to the actually pertinent question. If, as you agreed, your reformulated P1 is absurd - as it clearly is - going back to the original Colyvan version - if not even all scientific realists are ontologically committed to these "indispensable entities" - on what basis does the good Prof claim that "we ought to have" such "ontological commitment" a necessarily true statement? And if it is not a necessarily true statement - what business does it have masquerading as a premise in a syllogism? So with all due respect to my academic superiors - the Colyvan argument you presented is not a sound argument - regardless of how eminent its origins appear to be and whether I happen to use the correct terminology in a forum post or not. And more to the point, how can such an argument possibly lend credible support to your contention that scientific realism implies mathematical realism? It just doesn't does it?

No one argues that the thesis of mathematical realism is true because mathematical entities have volume and location in spacetime.

No one argues that anyone argues this - now who's inventing a confused straw man?

My argument is that an "entity", by definition, means something that has an independent existence - that is it is neither an invention of the mind (fictionalism), a linguistic element (nominalism) or an element of a more fundamental logic (logicism) (there are other ways of looking at math but these examples will do) and neither is it a mere property or attribute of an object to which we do have good reason to have ontological commitment (such as a particle, a galaxy or a universe) but which itself need not exist independently of the reality it is a property or attribute of.

Is E=mc2 (for example) an independently existing reality or a kind of story or concept that describes some aspect of the real universe or just a property of the matter and energy that do really exist or what??? And the answer is that we don't know. So on what basis do we refer to mathematical 'things' as "entities"? And without that word - what is the connection between the two premises? The argument seems also to have an unwritten premise that says that mathematical expressions are a kind of entity - i.e. that they have the kind of independent reality that other entities to which we (might) owe ontological commitment - such as particles...etc. have. If that is true, then mathematical things are a kind of entity and they are (unquestionably) indispensable to our best scientific theories and they (mathematical entities) are quite properly part of the correctly distributed middle in the argument. The argument remains valid but unsound. But if it is not true that mathematical things have independent existence, then their not being "entities" puts them outside the middle term "indispensable entities" because whilst they are indispensable they are not entities - they are, rather, fictions, names, formalisms or attributes - none of which (each for different reasons) are really entities. In that case, we don't even have a valid syllogism - let alone a sound argument.

the fact that mathematics is discovered, not invented, by humans is one of the primary premises by which to deduce the thesis of mathematical realism.

Well, that's a different argument - and one that I have not addressed so far. But does it really follow that discovery (as opposed to invention) entails reality? Presumably humans have also "discovered" that certain literary devices work better than others in fictional compositions - or that certain proportions of elements work better in works of art than others - but does that mean we have to have ontological commitment to the literary devices or the "golden section"? And in any case, how do we know that humans are the only entities capable of invention? Could math not equally be a story that the universe has invented about itself? Discovered by humans but no more fundamentally real than the fictions we create in our own linguistic symbolisms? And if not, why not?

 

[Nous]

Good Lord Nous! You are questioning my ability to reason logically? Here's what you wrote originally (verbatim):

Here's what I did (exactly what you said):

I don't know what you might have gotten in an email, but I assure you that at no point did I have an "etc." in any premise. And I think what I stated was not an AII-3, which, besides that questionable first premise, is the reason I deleted my reformulation. In any case, there is more than one way to make what Colyvan states as the Q-P indispensability argument into a valid argument. But what he states isn't valid.

I then noted that this was not a valid argument because P1 is not true - of course I should have said it is not a sound argument

Correct, you should have.

It is a valid argument. The subject is "scientific realists", (or "we" in Colyvan's) the predicate is their (our) "ontological commitment" and the middle is the "indispensable entities"

What the hell is that?

In the first place, “indispensable entities” is not a term anywhere, in anyone's syllogism. Other than that, I don't have a clue as to what type of syllogism you are saying Colyvan's statement of the QPIA is. State the S, P and M terms in order as you claim Colyvan's QPIA states.

As Colyvan states the QPIA, it isn't valid, especially as his terms lack quantification (except, strangely, for “entities that are indispensable”). What he literally states is this:

S are P.
M are P.
Therefore S are M.

That is not valid, besides lacking critical quantification. That form concludes this:

Rabbits are mammals.
Dogs are mammals.
Therefore, rabbits are dogs.

So now - to the actually pertinent question. If, as you agreed, your reformulated P1 is absurd - as it clearly is - going back to the original Colyvan version - if not even all scientific realists are ontologically committed to these "indispensable entities" - on what basis does the good Prof claim that "we ought to have" such "ontological commitment" a necessarily true statement?

I didn't mean that "not even all scientific realists are ontologically committed to the entities that are indispensable to our best scientific theories." As I quoted from the SEP on the other thread: “Scientific realism is a realism about whatever is described by our best scientific theories”. Scientific Realism (Stanford Encyclopedia of Philosophy) Thus, the scientific realist is someone who agrees with the thesis that whatever is described by our best scientific theories is real. No one can deny that mathematical relations (or structures) are described by by our best scientific theories.

I didn't like my premises about "scientific realists" because the essential argument is about metaphysical theses (i.e., scientific realism and mathematical realism), not about metaphysicians. The fact is that all philosophers et al. who might refer to themselves as "scientific realists" are not "ontologically committed" to any single thing.

No one argues that the thesis of mathematical realism is true because mathematical entities have volume and location in spacetime.

No one argues that anyone argues this - now who's inventing a confused straw man?

You haven't yet stated any fact by which to conclude that mathematical entities are not "ontologically real". One can easily get the impression that the reason you assert, but have not argued, that mathematical entities are not "ontologically real" is because they lack volume and/or spacetime locations.

My argument is that an "entity", by definition, means something that has an independent existence

What term would you like to use to refer to quantities, mathematical relations, functions, sets, ratios, predicates, geometrical objects (points, lines, topographies)?

Again, notice that neither of my arguments uses the term "mathematical entities". So since your "argument" is just an objection about the use of that term, it seems you have no objection to my 2 arguments.

Is E=mc2 (for example) an independently existing reality or a kind of story or concept that describes some aspect of the real universe or just a property of the matter and energy that do really exist or what???

E=mc2 is a mathematical relation that humans discovered to be a true statement about empirical reality. And to fill out the "whatever' in the SEP definition of scientific realism: "Scientific realism is a realism about the quantities and relations between those quantities described by our best scientific theories."

Well, that's a different argument - and one that I have not addressed so far. But does it really follow that discovery (as opposed to invention) entails reality?

Discovering relations that anyone using the same methodology can discover is discovering objective relations.

 

[siti]

OK - first of all, my sincere apologies for adding unnecessary confusion by unclear presentation and sloppy terminology.

Now, here's Colyvan's argument as you stated it:

(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities

We have both stuffed up our interpretations of it several times so I think "we ought" to tidy that up first...

What he literally states is this:

S are P.
M are P.
Therefore S are M.

That is not valid, besides lacking critical quantification. That form concludes this:

Rabbits are mammals.
Dogs are mammals.
Therefore, rabbits are dogs.

No - that's still wrong - what he has is this:

P are all M (I'm not sure that's the same as all P are M but because he also says "only" that implies that all M are P - he's just written it the wrong way round)
S are M (He doesn't state whether this is all or some but we can assume some I think - see below)
Therefore S are P (since we are assuming P2 means "some S are M" then the conclusion can only be "some S are P"

So we have an AII form (1 or 3 just depends on which way round we state P2 - either will do).

It is jumbled up but it is valid...to make it clearer I propose that we can rearrange P1 to say "all and only the entities that are indispensable to our best scientific theories are 'things' that we ought to have ontological commitment to" - that is really what Colyvan wants to say (yes?)

Then P2 is more or less OK if we put "some" at the beginning (because, presumably we agree that not all mathematical entities are indispensable to our best scientific theories), and omitting the "and only" bit from P1 (because it doesn't help the argument at all) so we now have a valid AII-1 syllogism (which is what you said you were aiming for? We could equally switch S and M in P2 and get a valid AII-3 form):

P1: All M [entities that are indispensable to our best scientific theories] are P ['things' we ought to have ontological commitment to]
P2: Some S [mathematical entities] are M [entities that are indispensable to our best scientific theories]
C: Therefore some S [mathematical entities] are P ['things' we ought to have ontological commitment to]

So Colyvan's IS a valid syllogism - just a bit jumbled in the form that you (he?) presented. And it also reaches the conclusion you (and he?) were aiming for - namely, to prove that (at least some) mathematical entities are real.

But it is still NOT a sound argument because there is no way to establish the truth of P1 - why "ought" we to have such ontological commitment?

And I still have a problem with the use of 'entities'. My understanding and explanation of that is fine in my previous post(s) as far as I can see (he said, marking his own work). There is absolutely no sound reason to assume that mathematical statements/relations...etc. exist as 'entities' at this point in the argument because we have not yet established that "we ought to have ontological commitment" to them and to say they are 'entities' is to have an ontological commitment to them. Of course it might be true that they are, but again, this is not sound argumentation (IMO) because it appears (to me at least) to assume the conclusion.

Well - it does, in fact, assume the conclusion and that can clearly be seen by attempting to substitute the word "entities" with another word like, say, "statement" or "relation". Presumably you have no issue with that because presumably you accept that some mathematical "entities" are "statements" or "relations"? But if one makes that substitution then it is very obvious that mathematical realism is assumed in the premises...this is not (as you suspected) a straw man fallacy, it is, I suppose, a kind of reductio ad absurdum:

P1: All M [statements/relations that are indispensable to our best scientific theories] are P ['things' we ought to have ontological commitment to]
P2: Some S [mathematical statements/relations] are M [statements/relations that are indispensable to our best scientific theories]
C: Therefore some S [mathematical statements/relations] are P ['things' we ought to have ontological commitment to]

On the plus side, we now have a much more precise and defensible P2, but P1 is even weaker than the original form and very obviously assumes the conclusion that some statements/relations are real. It becomes an absurdly circular argument - but all I have done is make what we mean by "mathematical entities" a little less obfuscated - n'est-ce pas?

Are you with me so far? If so - we can unwrap the other arguments more carefully one by one.

 

[siti]

OK - moving along to your two arguments and noting what you claim here:

So since your "argument" is just an objection about the use of that term, it seems you have no objection to my 2 arguments.

In this you refer to the fact that your two arguments do not use the term "mathematical entities" so I should have no objection on the grounds I have now clearly explained in my previous two posts. That is correct. But you have completely overlooked my objection to what you (may/may not) mean by "genuinely referential" in your first argument. Here is the argument as you stated it:

(1)

P1: All central terms of fundamental scientific laws are genuinely referential.
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are genuinely referential.
And I noted that it has the form AAI-3--which is:

All M are P.
All M are S.
Therefore, some S are P.

Again, as I noted above, you have a valid syllogism - I have never argued otherwise in this case.

And the argument is also sound. It proves that some "quantities/mathematical relations" (what are these if not "mathematical entities" if you are being consistent with you terminology?) are genuinely referential - meaning, I presumed in my previous post, that they refer to real entities. But how on earth does that make the "quantities/mathematical relations" real? How can this possibly be an argument for "mathematical realism"? Let's do a substitution thing again.

You agreed (I think) that my interpretation of "genuinely referential" was correct - i.e. that what you really mean is that "all central terms of fundamental scientific laws" are "pointing to things that really exist" - we could equally say then, I think, that they are "true statements about things that really exist"

If we substitute that phrase for "genuinely referential" then we get:

P1: All central terms of fundamental scientific laws are true statements about things that really exist
P2: All central terms of fundamental scientific laws are quantities (/mathematical relations).
C: Therefore, some quantities (/mathematical relations) are true statements about things that really exist

So when the obfuscation clears, we find that the argument really doesn't prove anything but what was already obvious. We are still no nearer a convincing (let alone compelling) argument for mathematical realism.