| | Sherman, you wrote, You imply that knowledge is not valid unless it is certain. I don’t agree. Man operates easily in everyday life using knowledge that is not certain. I do hold that knowledge is not valid (isn't knowledge) unless it is certain, but I wasn't aware that I implied this in anything I said. In any case, if a conclusion isn't certain, then it's not knowledge. For example, if I’m not certain that it’s going to rain tomorrow, do I know that it’s going to rain tomorrow. If I’m not certain of your age, do I know how old you are? No and no. A conclusion that constitutes knowledge must of necessity be a conclusion of which one is certain. What about probabilities? As I approach the next intersection, do I know that all traffic will stop when the light turns red? No; someone may run the red light. What I do know is that it is highly probable that all traffic will stop. And because I know this, my conclusion that it is highly probable that all traffic will stop is certain. Example for clarification of my meaning: He knows aspirin will relieve a headache but doesn’t know exactly why it works. So, he’s certain that it will relieve a headache, but he’s not certain how or why it works. This does not contradict the view that a conclusion which constitutes knowledge is certain. Your description of how we gain knowledge of the physical universe is basically correct, however the concepts "infinity" and "logic" are not strictly theories of the physical universe per se. One cannot gather empirical evidence by means of the scientific method to confirm or falsify them. Yes, one can. One can refer perceptually to those aspects of reality from which one acquires the concepts of “infinity” and “logic.” Without a direct perception of reality, one could not grasp these concepts.
I wrote, “Can one observe infinity empirically? Not directly as one would a physical object, because mathematically, it refers to the absence of any limit on the potential to extend a given size or quantity. But the concepts required for an understanding of infinity were arrived at by observing reality, i.e., by observing concrete instances of quantity, extension and limits. I don’t think you’ve thought this through clearly. Are you suggesting that man could not have knowledge of "infinity" prior to its use as a mathematical concept? I’m saying that there is no such thing as actual infinity. The only valid concept of “infinity” is the one pertaining to the potential for extending a given size or quantity. Early man could gaze at the night sky or at the distant horizon and observe an endlessness that he could conceptualize as "infinity." Aren’t you contradicting yourself when you say that early man could gaze at the night sky or at the distant horizon and observe an endlessness? If he can observe endlessness by gazing at the night sky, then, contrary to your earlier statement, endlessness or infinity is empirically arrived at. But is this true? Can one observe endlessness or infinity? The answer is, no.
To observe an object is to observe it in relation to that from which it is distinguished, i.e., to observe its limits. In gazing at the night sky, one observes celestial bodies, aircraft and weather balloons. The surrounding blackness is simply the absence of any discernible objects; it is not itself an object.
It is true that in observing that the horizon continues to advance as one moves along the earth's surface, one is observing the extension of a limit, an observation which could have given rise to the mathematical idea of a limit's being extended indefinitely. But the mathematical concept of infinity is valid only insofar as it refers to the potential for a process of extension or addition that does not have a limit beyond which it cannot take place. One can, in principle, always add another unit or degree to any given quantity. One can always count further than one has already counted. But however far one counts, the number to which one has counted will be finite. Infinity can never be actualized, because an infinite quantity is a contradiction in terms. A quantity, by definition, is some specific number, but an infinite quantity is no specific number. Furthermore, what do you mean by "concrete instances?" How do you observe concrete instances of "quantity, extension and limits?" If I observe three ducks, three cars, three trees and three fingers, am I observing the quantity three? I think not. Not as an abstraction, but you are observing four different groups of three objects and can see that each group has the same number of objects. To arrive at the abstract quantity “three” from that observation, however, you must mentally isolate the quantity that the four groups of objects share, ignoring their differences and focusing only on their quantitative similarity. The concept "quantity" (as well as the concepts "extension" and "limits") cannot be observed. It is a product of something else. The concept “quantity” cannot be observed. What can be observed are objects of a certain quantity. The concept “extension” cannot be observed; what can be observed are objects of a certain size and dimension. The concept “limit” cannot be observed. What can be observed are objects with limits.
I wrote, “Logic, as I said, is a concept arrived at by observing that every existent is what it is and not something else.” But you see, you haven’t observed a concrete instance of logic. You’ve "arrived at" the concept of logic by using logic ("every existent is what it is and not something else"). When man observes a duck, a car, a tree and a finger, he does not observe the law of identity. He reasons that there are existents of duck, car, tree and finger. He assumes every "existent is what it is." From this premise he reasons that there is a law of identity. True, he doesn’t observe the law of identity; he observes and identifies particular existents, from which he then grasps the law of identity through a process of abstraction. He doesn’t ”assume” that every existent is what it is; he sees that every existent is something in particular, and from that perceptual awareness, he is able to form the concept of identity.
I wrote, “To be sure, mathematical infinity and logic are abstractions from abstractions, but all abstractions are ultimately based on an observation of concrete reality.” But I’ve just shown that this is not true. Mathematics and logic have no concrete referents in reality that can be empirically observed. Yes, they do. The concept “three” refers to every group of three objects – just as the concept “man” refers to every human being. Similarly, the law of “identity” has as its referent the identity of each and every existent. They are products of human reasoning. The law of identity is recognized by means of human reason, but exists independently of it – just as the law of gravity is recognized by means of human reason, but exists independently of it. Logic is ontological, because identity is existential. Yes, the premises upon which they are based may have been inspired by concrete objects, but these premises were certainly not observed. What premises are those? I would go so far as to say that mathematics and logic are not only products of human reasoning, they are products of the logical structure of the human mind which exists prior to human observation of the physical world. A man can imagine the concept of mathematical infinity but not actual infinity because the logical structure of the human mind won’t allow him to imagine actual infinity, just as a man’s mind can’t comprehend a world in which the law of identity does not apply. I don’t know what you mean by “logical structure of the human mind.” The human mind is simply aware of existence as it is; there is no innate "logical structure" filtering that awareness. Such an idea would imply that one’s awareness is colored by the mind's logical structure and is therefore not a direct, unbiased recognition of reality. It would imply that one is not observing the world as it actually is, but only as one’s mind allows one to observe it. The logical structure, if you want to use that term, is in the object of awareness, not in the awareness itself; logic is fundamentally metaphysical, not epistemological. One cannot properly imagine actual infinity or a world in which the law of identity does not apply, not because the logical structure of the human mind won’t “allow” it, but because reality won’t allow it. The knowledge that A is A is not gained by man observing reality. The knowledge that A is A is not knowledge at all, but an immutable fact of man’s very nature, an undeniable result of the way man’s brain is wired. Not true, for the reasons previously mentioned. If one’s knowledge of identity were conditioned by the way that man’s brain is "wired," then one couldn’t know that identity is an immutable fact of reality, because one's concept of identity would pertain not to reality but simply to the way that one's brain is wired to perceive reality, in which case, one couldn’t possess knowledge of any kind.
I wrote, “Higher level abstractions are based on lower level ones, with the lowest level abstractions based on perceptual concretes. There is nothing else from which to form abstractions. So all knowledge is ultimately based on the evidence of the senses. Consider the concept 'chair,' which refers to every chair, past, present and future. A chair is a perceptual object, but the concept 'chair' is not a perceptual object; it is a mental integration of two or more chairs which, despite their individual differences, have the same distinguishing characteristics within the wider classification of 'furniture.' The same is true of the concept 'table.'
“But whereas the concepts 'chair' and 'table' are abstractions that subsume individual chairs and tables, the concept 'furniture' is an abstraction that subsumes the concepts 'chair' and 'table.' 'Furniture' is a classification of particular kinds of furniture, of which the concepts 'chair' and 'table' are particular units, just as individual chairs and tables are units of the concepts 'chair' and 'table'. And the concept 'furniture' is itself a unit of the wider concept 'household objects,' which subsumes other units such as art work, appliances and electronic equipment. But however far one goes up the conceptual ladder, the structure of knowledge must always rest firmly on the ground of concrete reality.” If all this is true, certainly you could name the "perceptual concrete" which is the lowest rung on the "conceptual ladder" with regard to mathematics and logic, i.e., the perceptual concretes on which mathematics and logic are based respectively. And I can. Mathematics is based on a perception of quantity in the real world (such as two apples, three oranges, five pencils, etc.), without which it couldn’t be conceived, and logic is based on a perception of existents in the real world, (such as entities, attributes or actions), without which it couldn’t be conceived. As I mentioned, your explanations of how we use the scientific method to falsify our theories about our physical universe is accurate, so there is no reason for me to rehash it. However, I do take exception to one thing you said: Theories can be falsified by inductive research, but theories can also be verified by inductive research. For example, the theory that (non-amorphous) ice floats in water can be verified by observing that the structure of ice takes up more volume than water molecules and therefore is less dense than water. Theories about the physical universe cannot by verified by inductive research…ever! You can’t have it both ways and say that sometimes they can. It depends on the theory. The observation that "ice floats in water" is not "inductive research." It is an observation that "ice" and "water" are two different things, two different concepts, two different existents. It’s not just an observation that they’re two different things; it’s an observation that they act in a certain way with respect to each other, and that mode of action is verified by observation, inductively, as it were. One arrives at the theory that ice floats in water by observing the nature of ice and water and their interaction, and then generalizing from that observation. Now, one may theorize that ice does not float on water, but such a theory is easily falsified by a single experiment. One might also theorize that ice floats on water and spend the next hundred years trying to falsify such a theory, but to what avail? Exactly, because it’s already been verified, inductively! It would be like trying to falsify by means of inductive research a theory that says a table is a chair. True, which is why Objectivism says that the law of causality is the law of identity applied to action. How a thing acts is a function of what it is.
To be continued...
- Bill
(Edited by William Dwyer on 12/10, 9:46am)
(Edited by William Dwyer on 12/10, 6:15pm)
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