James Still

<strong>I'm not saying anything about the properties of abstract entities other than that they are logically necessary (one need not be Platonic to hold to that).</strong>

If abstract entities are logically necessary then they must exist in every logically possible world, even those worlds in which human beings never evolved. That makes you a Platonist (a realist) with respect to abstract entities. You are saying that the abstract entities are ontologically prior to we humans who have "discovered" them. This assumption comes out of the blue with no explanation other than you "maintain" that it must be so. We could take you seriously if you offered an argument rather than your personal assurance that abstract entities are logically necessary.

geoff

James Still wrote: "You are saying that the abstract entities are ontologically prior to we humans who have "discovered" them. This assumption comes out of the blue with no explanation other than you "maintain" that it must be so. We could take you seriously if you offered an argument rather than your personal assurance that abstract entities are logically necessary."

Vorkosigan wrote: "That's fine that you "maintain" this, Geoff, but what everyone wants to see is a demonstration of why abstract entities are logically necessary AND cannot be dependent on contingent beings. Both forks of your assertion require proof. That would include a definition of "logically necessary," and an argument showing how it applies to the cognitive processes of H. sapiens."

My reply: In the narrowest sense, what is logically necessary is what follows from the laws of logic alone. Thus, a statement like 'either it will rain or it will not rain' expresses a logically necessary truth, because it is an instance of the law of excluded middle. Again, 'If all men are mortal and Socrates is a man, then Socrates is mortal' expresses a logically necessary truth, because in standard logic if we may deduce the consequent of a conditional from its antecedent, then the truth of the conditional follows from the laws of logic alone.

A sentence expressing a logically necessary truth, in this narrow sense, is true solely in virtue of its logical form: the meanings of any non-logical terms which it contains are irrelevant to its status as expressing a logical necessity. Thus, 'Either it will rain or it will not rain' expresses a logically necessary truth because it has the logical form 'Either p or not p'. However, in a wider sense a sentence may be said to express a logical necessity if, although not itself a sentence true solely in virtue of its logical form, it may be transformed into such a sentence by replacing certain terms in it by other, definitionally equivalent terms. Thus, various laws of logic are necessary. If humans are contingent beings, and contingent beings by definition might not have existed, how can necessary laws of logic and what follows from these laws be said to be dependent upon humans?

For those who remain unconvinced, here's a slightly different approach you may wish to respond to:

Numbers seem to be dependent upon intellectual activity. If there were no minds, there would be no numbers. However, there are too many of them for them to arise as a result of human intellectual activity. If numbers are dependent upon intellectual activity, and N&E necessarily excludes the possibility of an unlimited mind but affirms the existence of numbers as dependent upon human intellectual activity, then how can N&E affirm that n1...n2...ad infinitum exist? Put simply, the human mind cannot (or has not) thought of all the possible numbers. But they exist. How is that possible given N&E?

[ May 24, 2002: Message edited by: geoff ]

[ May 24, 2002: Message edited by: geoff ]</p>

Kyle Smyth

Geoff-

I am going to have to say that your OP is premised on either a) a far to shallow reading of the source material or b) a miss reading of the source material. Now I don't know how much you read of the document in question, but Mr. Augustine went in to great depth on your very question. His short answer (paraphrased) numbers (and other abstracts) are only logically necessary to the extent that we need them to interpret the universe.

"In any case, I will confine our exploration of the controversy over abstract objects to paradigm cases of abstract objects like numbers where the traditional definition of abstract objects does apply. There is nothing we can point to within space and time and say 'that is the number 4'. Furthermore, numbers and the relations between them are unchanging and mathematical truths like 2+2=4 seem timelessly true. Physical objects such as acorns can be arranged such that we can say that there are only four of those objects within a given space, but these objects exemplify instances of the number 4--they are not equivalent to '4' itself. On a Platonic account, four acorns are a concrete and particular exemplification of this abstract and universal form. So 4 is a universal concept rather than a particular one. The
number 4 is also an abstract concept rather than a concrete one, unlike the idea of an acorn. We cannot point to the number 4 in the way we can point to an acorn--this is the essence of what being an abstract object is.

Does naturalism allow the existence of abstract objects? Alan Lacey thinks that naturalism construes the natural world as a closed system of natural causes and effects "without having to accommodate strange entities like non-natural values or substantive abstract universals" (Lacey 1995, p. 604). Similarly, Arthur C. Danto thinks that naturalism entails the denial of the existence of abstract objects. Danto argues that formal sciences like mathematics
no more entail a Platonistic ontology than [the empirical sciences do], nor are we, in using algorithms, committed to the existence of numerical entities as nonnatural objects. If the formal sciences are about anything, it will at least not be a realm of timeless numerical essences, and at any rate logic and mathematics are properly appreciated in terms not of subject matter but of function, as instruments for coping with this world rather than as descriptions of another one (Danto 1972, p. 449)." K.Augustine

If no one ever thought of the number one, that would not mean that all of creation would 'pfffffffffttt----pop' in a logical contradiction.

[ May 24, 2002: Message edited by: Kyle Smyth ]</p>

owleye

geoff...

One interpretation of your dismissive reply suggests you have been given an assignment to justify a Platonic position with respect to abstract objects and for some reason you think we should help you with it. Another interpretation is that you wish to engage only those respondents who accept that premise.

No matter. My interest in the topic is not such as it needs to fall in line with yours.

owleye

Philosoft

What do you mean, "there are too many [numbers]"? Numbers are not physical things.

Numbers are most basically human shortcuts. Instead of saying, "a mouse, a mouse and a mouse," we can say, "three mice" and convey the same information. Fractions and decimals are just interpolations based on rules. Whole numbers are of utmost importance; we can essentially call the stuff in between whole numbers anything we want. Fractions of matter don't actually exist.

Vorkosigan

My reply: In the narrowest sense, what is logically necessary is what follows from the laws of logic alone.

Hi Geoff! The Primer I referenced above shows how the laws of logic arose through evolutionary processes, as a response to and a condition for human social interaction. The laws of logic are just shorthand descriptions of the way different bits of reality relate to each other.

However, in a wider sense a sentence may be said to express a logical necessity if, although not itself a sentence true solely in virtue of its logical form, it may be transformed into such a sentence by replacing certain terms in it by other, definitionally equivalent terms. Thus, various laws of logic are necessary.

"Necessary" to who? For what? "Necessary" is a subjective term.

If humans are contingent beings, and contingent beings by definition might not have existed, how can necessary laws of logic and what follows from these laws be said to be dependent upon humans?

What do you mean "dependent?" As far I know, logical abilities have been shown in many different animals, including mammals, birds and spiders. Jumping spiders of the genus Portia show serious abilities to calculate, experiment and plan.

Numbers seem to be dependent upon intellectual activity. If there were no minds, there would be no numbers. However, there are too many of them for them to arise as a result of human intellectual activity.

"Too many?" Huh? In order to generate a new number, all I have to do is add '1' to the biggest number I can currently think of. What's so difficult about that?

If numbers are dependent upon intellectual activity, and N&E necessarily excludes the possibility of an unlimited mind but affirms the existence of numbers as dependent upon human intellectual activity, then how can N&E affirm that n1...n2...ad infinitum exist? Put simply, the human mind cannot (or has not) thought of all the possible numbers. But they exist. How is that possible given N&E?

You're engaging in the philosophical fallacy known as equivocating. You've mistaken understanding infinity as a concept for the actual ability to hold an infinite number of numbers in our head. The two are not the same. I don't have to know every number in an infinite set to know that it is an infinite set. Heck, I know that by definition.

Put simply, the human mind cannot (or has not) thought of all the possible numbers. But they exist.

We haven't seen all possible stars, either, but they exist. How is that possible, given N & E?

Vorkosigan

HRG

First-order logic does not apply to the world, but to the language we use to describe the world. I have never seen a junctor connecting two real objects: it always connected two statements about real objects.
Easy. A world without sentient, intelligent beings - i.e. without minds to form those abstract concepts. The Solar system had nine planets (counting Pluto) before hominids became intelligent; but "Nine" as an abstract concept did not exist then.

See above.
&lt;snip&gt;
Exactly. The point is that they aren't "logically necessary". In zhe absence of intelligent entities, they cannot even be formulated.
"Logically necessary" - i.e. tautologous - itself is a concept which depends on the existence of intelligent entities.

Regards,
HRG.

Ierrellus

geoff,

Your example of logical necessity is given in one instance in the syllogism that states, "All men are mortal. Socrates is a man. Socrates is mortal." In the tightest abstraction of this syllogism we get if x=y and y=z, then x=z. If logic does not include in what ways x is equal to y, etc., one could conclude that all men are Socrates. An abstraction does not exist without reference.

Ierrellus

James Still

<strong>A sentence expressing a logically necessary truth, in this narrow sense, is true solely in virtue of its logical form: the meanings of any non-logical terms which it contains are irrelevant to its status as expressing a logical necessity. Thus, 'Either it will rain or it will not rain' expresses a logically necessary truth because it has the logical form 'Either p or not p'. However, in a wider sense a sentence may be said to express a logical necessity if, although not itself a sentence true solely in virtue of its logical form, it may be transformed into such a sentence by replacing certain terms in it by other, definitionally equivalent terms. Thus, various laws of logic are necessary.</strong>

At this point, of course, you are no longer speaking of abstract entities but instead tautologies in logic. Logic is a meta-language, a purely human construction that substitutes notation and signs for real-world entities in order to clarify and formalize their relations to each other. So we've shifted very far away from your initial concerns over metaphysical naturalism and abstract objects.

<strong>Numbers seem to be dependent upon intellectual activity. If there were no minds, there would be no numbers. However, there are too many of them for them to arise as a result of human intellectual activity. If numbers are dependent upon intellectual activity, and N&E necessarily excludes the possibility of an unlimited mind but affirms the existence of numbers as dependent upon human intellectual activity, then how can N&E affirm that n1...n2...ad infinitum exist? Put simply, the human mind cannot (or has not) thought of all the possible numbers. But they exist. How is that possible given N&E?</strong>

I agree that numbers are dependent upon intellectual activity (after all we constructed them in order to model the physical world). But when you speak of there being "too many of them to arise as a result of human intellectual activity" you have slipped right back into Platonism. The problem here is not with metaphysical naturalism but rather with your stubborn insistence upon treating numbers and other abstract objects as real beings.

Concerning the "laws" of logic, you might benefit from reading Wittgenstein. He said that philosophy was "a struggle against the bewitching of our minds by means of language." Then again I studied Wittgenstein extensively and he screwed me up for the rest of my life so perhaps you ought to think twice before reading him.

[ May 25, 2002: Message edited by: James Still ]</p>

Wizardry

I’m not asking you to establish the laws of logic. We agree that there are laws of logic. Our disagreement is about whether or not the laws of logic and other abstractions exist in a real sense outside the human mind. That is the only part of this discussion that has any repercussions at all for Metaphysical Naturalism.

The naturalist does not deny the laws of logic, but merely argues that they only exist as descriptions of the universe, not independently existing entities. You may not be able to concieve of a world where the laws of logic do not exist; the naturalist similarly finds it difficult to imagine a world where the laws of logic cannot be employed to accurately describe the world. The limitations of the human imagination does not supports neither argument.

You still have yet to establish that abstract entities exist independently of human cognitive activity. In fact, you still have not presented any argument other than “I can’t imagine it being any other way”, which is far from convincing.

If N&E is correct, then numbers are not abstract objects, so this entire argument fails.

Peace out.