I'm starting with this definition of yours. I like this definition of "objective facts."
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But it puzzles me why you can't see how math is objective.
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Yes or no, Mike-- is this true regardless of what anyone thinks?
The short answer is no. The Pythagorean theorem is not an objective phenomenon of mass/energy with spatio-temporal extension. It is an artifact of an analytic abstract system that was created by human brains and is brain dependent for its existence. The theorem is always provably true within the rules of the Domain, which would be no different from things provably true in other analytic Domains that consist of a different set of rules. Any abstract analytic system requires *someone* to *think* up the rules in the first place.
Does our (humanity’s) math exist if there are no longer any human beings? Can we presume our abstract quantitative and computational language we call mathematics, which was created within the performance limits of our biological brains, is the only possible way of thinking quantitatively about the Cosmos? Can we even imagine a quantitative and computational system incompatible with the physical limitations of our biological computing capacity?
The Pythagorean theorem is an artifact made possible by the rules of the analytic system we have created. We can mentally overlay the abstract conception of a triangle onto things in the real world, and with our abstract quantitative language, make useful quantitative conclusions within the limits of our ability to measure. Our imagined triangle overlay isn’t objectively real, just the objects or phenomena that we overlay it upon.
What if we consider descriptive languages like English or Japanese instead. If different languages use different sets of verbal sounds to create words and use different rules for word order in sentence structure, might we imagine a rhyming scheme that works well for one language being incompatible with another? Is the rhyming scheme of the first language an objective fact or an analytic fact? Is it objective regardless of what anyone thinks, or is it an artifact of the abstract analytic language created in/by the human mind?
Yes, because all I'm saying is that math is just as objective as science.
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It deals strictly with things which are true whether anyone agrees with them.
Can math be used incorrectly? Can math be used correctly, like Logic, with a given set of presumptions, and be used to draw conclusions that are not possible in the real world? Two plus two equals four, by itself, is meaningless abstraction. Twos of what? Are the things being added identical in nature, compatible, or using the same unit of measure?
Math is abstraction and not objective in and of itself. It is simply a quantitative and computational language we may use to *describe* objective things in the world and great care must be taken if that is our intent. Math, however, doesn’t have to apply to the real world, the world of experience. Mathematics can simply be used to explore what is possible within the rules of mathematics itself, purely within the mathematical Domain. It can also be used to describe other imagined worlds that operate with a set of rules distinctly different from the actual world of our experience in the same way verbal language can.
The objective world is the world of phenomena consisting of mass/energy with spatio-temporal extension and events of interactions related to mass/energy. The subjective world is all that is in our heads. All human thought is subjective abstract representations. The abstractions do not literally share the properties of the things they are meant to represent and that is why, in our heads, a 400 lb pig can fly and we can imagine a creature with the body of a lion and head and wings of an eagle and call it a Griffin.
I am sure you have heard the expression, “A map is not the territory”, and it is apt to this discussion. Language, Mathematic, Logic are all abstract tools we can subjectively use for mapping/describing the objective world. Science, utilizing the principles and standard of scientific inquiry, is the demarcation tool we use to ensure our abstract representations, when used to describe the real world, the objective world, those abstractions being both of verbal language and quantitative language, actually does so to the greatest extent possible. Science, then, is the error detection and correction mechanism employed to verify the correspondence and utility of the *maps* we build of the objective world.
There are right triangle-ish things in nature. The more right triangle-ish they are, the more the pythagorean theorem describes their angles and area accurately. The less right triangle-ish the shape is, the less able the pythagorean theorem is to describe its properties. So what?
So, the triangle-ish things are the objective phenomena in the real world and the Pythagorean theorem is a subjective analytic fact in the subjective Domain of Mathematics. Would you consider the standard of dividing a circle into 360 equal degrees to be an objective fact?
Different cultures have created computational language using a different radix, or number of unique digits in their positional numeral systems. The decimal system is most widely used today, but other cultures have used a different base. Apparently the Maori used a undecimal system, languages in Nigeria used a duodecimal system, Basques and Celts used a vigesimal system, others have used ternary, quaternary, and quinary. Today our machines use a computational language based on binary and hexadecimal systems. These examples speak to me of the subjective analytic nature of mathematics.
And the same limitation of scope is present in science. Newtons laws of gravity can't be used to explain chemical reactions.
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Like the Pythagorean theorem, Newton's laws have their scope.
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Here is a great example of what I am saying. Mathematics, the analytic system, does not know what celestial bodies are. It does not know what mass is. It does not know what the speed of light is. It is simply a quantitative and computational abstract language. To describe our observations on motion we create abstract concepts such as Force, Mass, and Acceleration and explain their relationships with descriptive language. We can also represent the relationships in the mathematical language form of equations that allow us to either explore hypothetical quantitative value relationships or use actual observed and objective data of known variables to calculate an unknown variable. Analytic mathematical facts and rules of operations enable us to think and communicate subjectively about objective things in a synthetic way. We are able to create a useful subjective representation of what is objectively occurring.
Your definition here is incongruous with the other definition you gave. I don't like this one very much. Especially if you are wont to call science objective (which I think we both are).
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I didn’t like the sentence when I wrote it. It was not very clear. Here is my position. *All* mental activity is subjective. *Some* of that subjective mental activity is meant to describe objective phenomena in the real world in a synthetic and useful way. Scientific inquiry is a modality of inquiry and investigation that endeavors to verify, to the greatest extent possible, the degree to which our subjective descriptions of objective phenomena are synthetic and useful representations. I must emphasize again that since we are not omniscient as to all aspects of the real world, any subjective statements about the objective world are presented with varying degrees of confidence. Scientific statements are still only *maps* of the objective world, not the world itself, and much like ancient mariner maps with detailed drawing and descriptions of home coasts and progressively less detail further from home ending in uncharted waters, our “scientific maps” of the objective world form a similar pattern. Science, then, is used to police both our use of verbal language and mathematical language when our intent is to use either or both to describe the objective world.
This idea is nicely illustrated with the Newton example. Newtonian laws, the foundation of Classical Mechanics, gave a boost of confidence in the level of understanding of how and why celestial bodies moved in their respective patterns. Although it provided an improved perspective on that aspect of the objective world, from our further improved perspective of today, it was still an incomplete picture. General and Special Relativity have added some more detail to our *objective map* of the Cosmos, but we still recognize from this new vantage point that the map is not yet complete.
So, instead of saying, “Science is objective”, it seems better to say, “What is observed and verified by science is as objective as it currently can be”. All this is not to say that there are not aspects of our macroscopic world that we have complete confidence in such that our subjective statements are assumed to be strongly representative of the objective. These, as we discussed earlier, would be considered facts as described in your second definition of the word, or objective facts as opposed to analytic facts.
Given what I have said above, perhaps it would be better to substitute the phrase "synthetic facts" for "objective facts" in the interest of clarity and as an acknowledgement that any statement is made up of subjective abstractions. So, facts in your second definition would represent synthetic facts as opposed to analytic facts.
You can, but it would be a rather bloody affair.
It wouldn’t be sheep anymore though, it would be mutton.
A repeating decimal is merely the result of an inadequacy of the decimal system to express certain fractions. It's not a crime against nature.
Correct. Not a crime against nature, simply an artifact of a subjective analytic system.
As I said before, I take it as a given that math is thoroughly objective. If you aren't convinced of math's objectivity, it follows that you should neither find any argument for moral objectivity compelling. Are we at an impasse?
The abstraction of mathematical language, like verbal language, can be used to describe or convey information about the objective world. I don’t think we are at an impasse. All that is required is to demonstrate that our abstract constructs we call morals, or some aspect of them, describe or are derived from some objective phenomena in the real world, the world of experience. The stronger the evidence, the more confidence we will have in the objective status of morals.