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Your thoughts on Hume's case against induction?

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Not in that case.  "1 + 1 = 2" can only be applied to PART of reality, i.e., things which exist in discrete, similar, countable units.  Thus 1 apple plus 1 apple equals 2 apples, but what applies to apples doesn't apply to "justice" or "happiness."

Not really - "1 + 1 = 2" applies to relations of concepts, ie our concepts of number. Saying "1 + 1 = 2" isn't the same as saying 1 apple added to 1 apple gives 2 apples - theres a difference between grasping the truth of "1 + 1 = 2" in an abstract sense, and applying it to a material part of reality.

It applies to ANY part of reality -- but ONLY that part of reality -- that exists in discrete, similar, countable units. The units don't have to be material. If you have one objection to my ideas today and one objection tomorrow, that adds up to two objections.

In some cases adding 1 object to another object doesn't give 2 objects, for example adding a sugar cube to a cup of water,
That's because they are not in discrete, similar, countable units.

but this doesn't alter the fact that "1 + 1 = 2", always and forever.

But only in the context of discrete, similar, countable units.

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As I understand it, logical validity relates to the form of an argument, and providing a deductive argument follows the correct form, it is valid, so there’s no need for an appeal to empirical truths to establish validity.

Then how do you establish that one form of argument is correct and another form of argument is incorrect? By referencing and appealing to WHAT?

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Betsy: “Actually, I doubt you think ANY knowledge can ever be certain. Correct me if I'm wrong, but isn't your position that NO inductive generalization is EVER valid?”

No. I said induction cannot be logically validated, but inductive generalizations can be empirically justified.

How? :confused: Please explain in detail.

Since a claim that is contextually certain can in fact turn out to be mistaken, “contextually certain” in effect boils down to probability, so the distinction is just a matter of semantics.
No it doesn't. You can be certain if and when you have reduced your causal explanation or generalization to an identity -- at least in the part of reality (i.e., the context) in which it applies.

One might then ask what is the purpose of adopting a phrase such as contextual certainty rather than probability, and in my view it simply adds a veneer of certainty where none exists. And if ‘contextual certainty’ does equate to ‘certainty’, as you claim, why not, in the interests of economy, just drop the ‘contextual’?

We often do when the context is understood. If not, it must be specified. That way I can say, "I am certain that my conclusion applies to THIS part of reality (i.e., this context), but I do not have enough knowledge to make any claim, with certainty, about other parts of reality (other contexts).

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It applies to ANY part of reality -- but ONLY that part of reality -- that exists in discrete, similar, countable units. The units don't have to be material.  If you have one objection to my ideas today and one objection tomorrow, that adds up to two objections.

Are numbers, taken purely as concepts (ie the concept of 'one' as opposed to 'one apple'), single discrete countable units? When I assert that "1 + 1 = 2", I'm not making a statement about any aspect of observable reality in the sense of counting 2 individual objects, I'm asserting a truth about mathematical concepts and the relationships between them.

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Are numbers, taken purely as concepts (ie the concept of 'one' as opposed to 'one apple'), single discrete countable units?

They are units of the concepts of "concept," "integer," "quantity," etc. If I have "1" and "9" then I have two integers.

Unlike the Platonists and neo-Platonists, however, Objectivism holds that numbers are only concepts and have no existence separate from the entities they are used to count.

When I assert that "1 + 1 = 2", I'm not making a statement about any aspect of observable reality in the sense of counting 2 individual objects, I'm asserting a truth about mathematical concepts and the relationships between them.

Actually you are making a statement about relationships between any and all existents that exist in discrete, similar, countable units.

As an aside, this is an example of the proper way to do philosophy. No matter how abstract your ideas get, always seek to keep them tied to observable reality.

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...Since a claim that is contextually certain can in fact turn out to be mistaken, “contextually certain” in effect boils down to probability, so the distinction is just a matter of semantics.

What is the probability - or the basis for any doubt whatever - that the earth orbits the sun, that a cow can't jump over the moon, or that the WTC was not destroyed and is still standing?

Just semantics?

Look, with the exception of knowledge which pertains to existence and our knowledge of it as such (axioms), all knowledge is contextual. All of it. No exceptions. What you are saying in effect is that since we are not omniscient and can't know everything, we can't know anything.

The irony is that the position you are upholding claims to be empirical and yet you deny the most evident fact of human knowledge, that we are absolutely certain of a great deal and that our knowledge is continually growing by leaps and bounds. How do you think civilization was built and that we got to the moon and back? Was it luck, chance, a miracle perhaps?

Fred Weiss

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Concepts have to be formed over instances from reality. If so, then how can Hume's concept of knowledge be valid, when it requires omniscience? Where did he get these instances of omniscient entities to form his concept of knowledge over? From religion.

He is one of these guys who thinks they have escaped religious influence but are

really still in it's grasp.

:confused: Ninjas are more powerful than Hume :ph34r:

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They are units of the concepts of "concept," "integer," "quantity," etc.  If I have "1" and "9" then I have two integers. 

Unlike the Platonists and neo-Platonists, however, Objectivism holds that numbers are only concepts and have no existence separate from the entities they are used to count.

Well that's fine, its essentially what I would say too. However that still allows us to talk about '1' and '2' independently of there being one or two objects (I'm not sure my terminology here is correct, but we would be talking about '1' and '2' as units of the concept of number rather than about '2 apples' or whatever).

Actually you are making a statement about relationships between any and all existents that exist in discrete, similar, countable units.

No, I am making a statement about '1' and '2', completely divorced from any particular existents. As an example, consider a more complex mathematical expression such as "e^iπ + 1 = 0"; what existents are being referred to here?

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No, I am making a statement about '1' and '2', completely divorced from any particular existents. As an example, consider a more complex mathematical expression such as "e^iπ + 1 = 0"; what existents are being referred to here?

n = pi, the ratio of the circumference to the diameter of a circle.

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Im confused, are you talking to me here or clarifying it for other people?

I gave you an explicit answer to your question. You asked:

"As an example, consider a more complex mathematical expression such as "e^iπ + 1 = 0"; what existents are being referred to here?"

The solution to the equation is n = pi, and pi is the existent that is the ratio of the circumference to the diameter of a circle.

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I gave you an explicit answer to your question.

Oh, I get what you mean. I didnt intend 'n' as a variable, I meant it to represent pi (it's some bizarro character I found in character map that looked like pi. I admit that it is almost indistinguishable from a lower case N however).

I was asking what specific existants in reality were being mentioned in the identity"e^((pi)*i) + 1 = 0" to demonstrate what I mean by taking numbers as purely seperate from any existing objects. The identity isnt claiming that if you take 'e' bananas and raise them to the power of "pi * the positive root of minus one" then you will be left with -1 bananas - it is asserting a fact about numbers and concepts.

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No, I am making a statement about '1' and '2', completely divorced from any particular existents.

But if those concepts aren't ultimately reducible back to your sense data, then they are invalid concepts. They are, however, in fact reducible, back through the concept unit and then backwards into perception. How do you know 1 + 1 = 2, whatever enabled you to make that claim if not some abstraction from reality.

On a separate note, for Betsy, I'm interested in how you deal with the problem of induction. It makes a lot of sense to me, is this idea also talked about in Peikoff's recent lectures on induction (which I intend to buy soon to help me think about induction more).

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I was asking what specific existants in reality were being mentioned in the identity"e^((pi)*i) + 1 = 0" to demonstrate what I mean by taking numbers as purely seperate from any existing objects. The identity isnt claiming that if you take 'e' bananas and raise them to the power of "pi * the positive root of minus one" then you will be left with -1 bananas - it is asserting a fact about numbers and concepts.

But those numbers and concepts are, ultimately, reducible to perceptual reality. You should really read that book in detail, ITOE. You know the one, the one that you think is "a bit shallow." We build abstractions from abstractions, to ever-higher levels of abstraction, but if we cannot reduce those abstractions to perceptual data, then we cannot really justify our knowledge as being connected to reality. Hence, the rationalism seen in most all of modern philosophy.

The formula e^(i*pi) +1 = 0 is just a polar representation in the complex plane that can be perceptually visualized by projection onto the Riemann sphere. It represents the simple statement cos(i*pi) + i*sin(i*pi) = -1, a particular radius vector. Each of the elements, e, i, pi, 1, 0, are either directly reducible to perceptual reality or are concepts of method reducible to a purpose which itself is reducible to perceptual data.

The only issue here is the extent of your knowledge and how that knowledge was acquired and how it is held in your mind. If you cannot reduce higher-level abstractions that you hold, to perceptual reality, then you cannot verify the hierarchical struncture of your knowledge and you cannot be sure that what you hold as knowledge is connected to reality.

Maybe it is not Ayn Rand who is "a bit shallow." :confused:

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But if those concepts aren't ultimately reducible back to your sense data, then they are invalid concepts.  They are, however, in fact reducible, back through the concept unit and then backwards into perception.  How do you know 1 + 1 = 2, whatever enabled you to make that claim if not some abstraction from reality

I'm not sure what 'reducible' means in this context. I could give you an example of 2 objects, but I couldnt give you an example of half an object, -3 objects, or 'i' objects. Also, I can freely use the number '3490324738032432' and 'understand' it just as well as I 'understand' the number '3', but I doubt I've perceived that many objects in my entire life.

Obviously we learn the natural numbers through perceiving objects and abstracting ('2' is what two apples and two oranges have in common etc), but once we've formed the abstract concept there is no need to reduce it back to sense data, and it is possible to talk about '2' as a concept without mentioning "two OF something'. When I say '2', I dont mean 2 apples, pigs or oranges. I just mean '2'.

(In the same way, people generally get the idea of 'pi' from dealing with circles, but they will often go on to use pi in other contexts that have very little to do with circles. To keep thinking 'circles!!' whenever you encounter pi would be as silly as thinking 'triangles!!' whenever you encountered things relating to sines and cosines).

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But those numbers and concepts are, ultimately, reducible to perceptual reality
Again, I'm not completely sure what is meant by reducible here. When I talk about the concept of 'pig' I can reduce this to perceptual data, namely the multiple pigs that I have encountered in my life. When I talk about 'unicorns' I can reduce this to perceptual data in the sense of pictures of unicorns that I have seen, and also to the simpler components of 'horn' and 'horse' which are combined in the whole. However, I am not sure how things such as real numbers, negative numbers, 'raising to the power' and so on are reducible in this way. Bear in mind that my original point was that we are able to talk about numbers simply as numbers, not as numbers OF things (ie '2' opposed to '2 apples'). In order to form a hierachy including concepts such as negative numbers, we first have to isolate the concept of number. We cannot talk about "-2" of anything.

Maybe it is not Ayn Rand who is "a bit shallow."  :confused:

Are you intending on spreading the personal attacks over multiple threads, or is 2 going to be sufficient? I already answered your ad homs in thread where you first made them.

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Again, I'm not completely sure what is meant by reducible here.

Then you should learn. You can start by reading Ayn Rand's ITOE. You know the one I mean, the one you thought was "a bit shallow."

Are you intending on spreading the personal attacks over multiple threads, or is 2 going to be sufficient?
Whatever it takes to reduce your remarks to the ignorance of the position from which they were made.

I already answered your ad homs in thread where you first made them.

Never assume I have read what you write.

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No, I am making a statement about '1' and '2', completely divorced from any particular existents.

You can't divorce numbers from any particular existents, because it is only particular existents that exist in discrete, similar, countable units that give any meaning to the numbers. They are the logical roots of any concept of number and they come along with the concept whether anyone acknowledges it or not.

Without those existents that exist in discrete, similar, countable units, numbers are just stolen concepts or floating abstractions.

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You can't divorce numbers from any particular existents

I'm not sure why. I can divorce 'red' from any particular existant - obviously this doesnt mean that theres some kind of abstract 'redness' floating around in a shadowy platonic realm. It just means that I can talk and think about 'red' as a concept itself, rather than just 'things that are red'. My concept of red has obviously been derived from perceptual data, just as a child's idea of '2' has.

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On a separate note, for Betsy, I'm interested in how you deal with the problem of induction.  It makes a lot of sense to me, is this idea also talked about in Peikoff's recent lectures on induction (which I intend to buy soon to help me think about induction more).

I have been thinking about induction and writing about it for the past seven or eight years and have definite ideas about the subject. I took Peikoff's course in person at the two summer conferences in which he gave them. Peikoff takes a very different approach than I do.

To clarify the matter in my own mind, I am currently discussing induction by e-mail with some professional Objectivist philosophers. I am also doing more writing on the developmental aspects of induction and how children form their first inductions. That is an area that Peikoff has a lot to say about that I haven't paid much attention to until now.

I'll be glad to discuss my own theories of induction, but I'd rather hold off discussing Peikoff's views until I have a better understanding of them.

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I can divorce 'red' from any particular existant - obviously this doesnt mean that there's some kind of abstract 'redness' floating around in a shadowy platonic realm. It just means that I can talk and think about 'red' as a concept itself, rather than just 'things that are red'. My concept of red has obviously been derived from perceptual data, just as a child's idea of '2' has.

If you are proceeding properly, you are abstracting the characteristic of redness from the existents that possess the characteristic. You should not divorce the characteristic from the existents. If you do that, you lose the concept's tie to reality.

Maybe that's how people end up with floating abstractions. They form their concepts normally and later forget where their concepts came from and what they mean.

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DavidOdden: “In fact, that is a fundamental statement of logic in Objectivism: conclusions must be derived by applying logic to true statements…I'm aware of the existence of types of formal logic that would label such a derivation "valid" and I grant you that this is the most common construal of the concept in academic studies on logic, it is not how ordinary logic works and is also is not the only variety of academic logic on the market.”

All that is true, and that’s what this thread is about. As I say, I am talking about standard formal and informal logic, where the standard understanding is that different considerations apply.

DavidOdden: “Setting aside the fact that all mice aren't black, you cannot stipulate that statement ex nihilo -- it is not axiomatically true. You cannot arbitrarily introduce an untrue statement and derive anything valid. So you would first have to prove the universally quantified claim, which has to be done inductively.”

I’m not claiming axiomatic status for the statement. I guess we have a different view of deductive arguments. In my view, one of their major functions is to draw out or highlight the implications of the major premise. In that case, conflating validity and truth can lead one to assume that since the form of the argument is valid, the content is also true, when this is not necessarily the case. Counter-examples, such as my black mice one, highlight this aspect. Also, since the example cannot be faulted on form, this at least shows that a distinction between form and content can in fact be made.

DavidOdden: The expression "contextual certainty" is redundant: simply saying "certainty" suffices because the basis of certainty (knowledge) is always contextual…Once you have dealt with that fact that seemed to cast doubt -- and therefore there remains no doubt -- then you are certain.”

Yes, but at what point do we know that all the facts are in? We know from history that our knowledge on all sorts of subjects changes, sometimes quite abruptly, and in a way that invalidates previous knowledge. (Of course, this is also an inductive argument, but its justification lies not in appeals to the past, nor to some supposed “nature” of the scientific enterprise, but rather in future outcomes.)

E

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