posted
I believe that "objective truth" is a useful concept regardless of whether it actually exists (I feel the same way about God too).

I do not believe that it is possible for any human to actually know the "objective truth". Humans who hoenstly claim to know the objective truth are either fools or prophets (or both). One of the tenets of my religion is that these days it is impossible to know the difference between fools and prophets (No offense meant to any Mormons, I do respect your religion and I hope you can respect mine).

I consider people who think that their understanding of "objective truth" is better than that of others to be extremely arrogant and I'll be the first to acknowledge that I can be extremely arrogant about some things.

For many centuries people believed that "the sum of the angles in a triangle is 180 degrees" is an objective truth. Most rational and informed scientists today no longer consider that statement to be an objective truth (it is false in non-Euclidian geometries). Euclidian geometry is such a useful model that it is still tempting to treat it as an "objective truth" even when it is not.

I believe that 1+1=2 in most contexts but I recognize that there are contexts when 1+1=1 or 1+1>2 are much more useful paradigms.
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posted
WE can't get it. Like what Pete told me, how when we're having sex we're actually masturbating.
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posted
"For many centuries people believed that "the sum of the angles in a triangle is 180 degrees" is an objective truth. Most rational and informed scientists today no longer consider that statement to be an objective truth (it is false in non-Euclidian geometries). Euclidian geometry is such a useful model that it is still tempting to treat it as an "objective truth" even when it is not."

Both Euclidian and non-Euclidian are OBJECTS of human thought. So "the sum of the angles in a triangle is 180 degrees" is objectively true within the object to which it is subject, Euclidian geometry.

I think the philosophers and epistemologists trip over themselves a lot (which is why their tenets change so much in an almost faddish fashion).

I think a much better term for what is called "objective truth" would be 'universal truth': a truth that is true any and everywhere, regardless.

One could see that term and realize that since the human mind is incapable of perceiving the totality of the universe, such a concept remains only a concept.

But in the realm of consensually mediated reality, 1+1=2 in a manner invulnerable to subjective input.

One must change the definition of 'one' to change that axiom. One must alter the object. Altering objects will of course alter those objects' truth correspondingly.

But, for the record:

quote: In philosophy, an objective fact means a truth that remains true everywhere, independently of human thought or feelings. For instance, it is true always and everywhere that 'in base 10, 2 plus 2 equals 4'. A subjective fact is one that is only true under certain conditions, at certain times, in certain places, or for certain people.

That's a wiki.

"I believe that 1+1=2 in most contexts but I recognize that there are contexts when 1+1=1 or 1+1>2 are much more useful paradigms."

I would call that a subjective belief.
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posted
Kenmeer makes the crucial distinction. An "objective" truth is not an absolute truth - it is contextual, consistent to all observers within a common external referent.
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posted
Just because a triangle on a sphere on some other surface doesn't have angles that sum to 180 degrees doesn't mean it isn't an objective truth in euclidean geometry. In mathematics you can clearly define what type of environment you are working in. In mathematics the base assumptions are called axioms and then everything that follows is objective truth based on those minimal assumptions. If you are interested about what can and cannot be determined objectively under different axioms read about Godel's incompleteness theorem.

Working with the natural numbers as normally defined 1+1=2 is an objective truth. Just because there exist systems where the result can be different doesn't make the statement any less objectively true under the normal assumptions/axioms.
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posted
As an old buddy of mine once said (who happens to be a math professor at a major university), "mathematics is a system of logic." Given a set of axioms, you can objectively "prove" certain propositions, which simply means the propositions are consistent with the given axioms. Nothing more.

The universe is not that simple. Although it appears to follow rigid rules (which explains why the hard sciences are so successful in predicting outcomes in many circumstances), often times it is not as exact in following the rules we have as we'd like.

For instance, the universe does not follow rules of calculus. Calculus posits that we can divide things into infinitely small segments, and any physicist can tell you that is impossible. You eventually get into quantum wierdness and the mathematics breaks down. But for macroscopic purposes, it works very well, and thus is accepted as "proof."

What this all means is that, as you state, you can't "prove" anything absolutely in this world. But that does not mean you can't "prove" something to an extent that is acceptable to a vast majority of observers. It may not be absolutely "true," but if it is practically true, that is about as good as we can get.

And that is enough for mere mortals.
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posted
"The universe is not that simple." Really? I'm not sure where to begin with this, but are you suggesting that there are no axioms the universe operates on or are you echoing seagul in: "I do not believe that it is possible for any human to actually know the "objective truth"." Frankly, I'm intrigued by the first option. Is it even possible? I don't think I can express my thoughts on this right now - maybe after I sleep. Hopefully more later.
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posted
In mathematical logic, axioms by themselves are not enough. You need to define a formal language before you can state the axioms. You also need to be given rules of inference for constructing true propositions.

For example, a formal logic system with only two axioms: 1. Politician lie. 2. Cast iron sinks. Is not enough to tell us if "politician lie and cast iron sinks" is a true proposition and how it differs from "politician lie in cast iron sinks". We need inference rules to tell us how manipulating statements using the words "and" and "in" affects their truth. We also need the formal language to resolve the ambiguity between using "sinks" as a noun or as a verb.

The choice of the Formal Language, the Axioms and the Rules of inference can be critical to what can or can not be proved.

When the field of non-Euclidian geometry found its first practical applications there were heated debates about whether the Axiom of parallel lines was an objective (or universal) truth. As time went on it became "obvious" that while Euclidian geometry remains as useful as ever, replacing the Parallel Postulate with "equally valid (though less intuitive)" axioms resulted in elliptic and hyperbolic geometries which are very useful as well. "An implication of Einstein's theory of general relativity is that Euclidean space is a good approximation to the properties of physical space only if the gravitational field is not too strong".

I have heard heated debates between mathematicians about whether the Axiom of Choice is true or not. Some of the them believed in the axiom of choice with a deep almost religious convictions and treated those who wanted to spend their time studying (and teaching) other formal systems as blasphemous heretics that were wasting the time and resources of their community. My impression was that alternate systems that do not include the axiom of choice do not yet have enough practical applications to earn the respect (and funding) that non-Euclidian geometry has but that it is possible (and according to some even likely) than it will do so in the future.

quote:in the realm of consensually mediated reality, 1+1=2 in a manner invulnerable to subjective input

Obviously if binary notation was used the same statement would be expressed as 1+1=10 but changing notation would have no effect on the truth of the statement. The following four statements: * 1+1=10 (in binary notation) * 1+1=2 (in any base greater than two). * one plus one equal two (in English) * uno más uno es igual a dos (in Spanish) All express the same concept which can be proved if you accept the peano axioms and the common definition for addition regardless of what language we choose to express them in.

But, no matter what language we write it in, the concept that "one plus one equals two" is not an objective truth in a world where the axioms of number theory are not true.

For a long time, even after I studied formal logic and understood the improtance of Non-Euclidian geometry, I still could not imagine a world in which those axioms would not be true. I was open to the idea that such models could exist but I could not concieve of one that had any meaning to me and at the time I would have agreed that 1+1=2 was an objective truth.

Then I mentioned this to someone who said these two simple sentences:

"One cloud plus One clound equals one cloud" "One man plus one Woman equals more than two people"

If you translate these two sentences back into mathematical notation we get: 1+1=1 and 1+1>2.

Since that day, I no longer consider 1+1=2 to be an absolute/universal/objective truth.

Don't get me wrong, I still love number theory and use it in the appropriate context. But I do not believe in it religiously and I think that my world is richer for being able to see other perspectives as well.

quote:Originally posted by seagull: Then I mentioned this to someone who said these two simple sentences:

"One cloud plus One clound equals one cloud" "One man plus one Woman equals more than two people"

If you translate these two sentences back into mathematical notation we get: 1+1=1 and 1+1>2.

Since that day, I no longer consider 1+1=2 to be an absolute/universal/objective truth.

Don't get me wrong, I still love number theory and use it in the appropriate context. But I do not believe in it religiously and I think that my world is richer for being able to see other perspectives as well.

The translation back to mathematical notation is wrong. A cloud is a collection of water particles. Meaning that to think about clouds in a mathematical setting you would need to think about a cloud as a set made up of all its particles. Then "adding" the two clouds would be taking the union of the two sets that make up each cloud.

The second statement is also difficult to translate back into a formal mathematical language. Words are a difficult thing to transform into logic. There is also a bit left out from the statement that makes the logical translation seem almost reasonable but it is really saying one man plus one woman plus a sperm meeting an egg is more than two people. So if you want to try to define what you mean by plus and how you want to count embryos then maybe you could take another shot.

It's fine to discuss philosophy in the vagaries of those two statements but it is meaningless and misleading pretend like there is some formal mathematical meaning in how you interpreted either one.

To respond specifically to your last statement number theory is a tool. If you have read about Godel you know that the theory of natural numbers is inherently incomplete. I'm not sure what is gained by making vague statements in English and "translating" them into the language of mathematics in an imprecise way. Mathematics is about precision and logic. I think your for your statements to have any meaning it should be viewed from a philosophical point of view not a mathematical one.
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quote:Originally posted by Wayward Son: For instance, the universe does not follow rules of calculus. Calculus posits that we can divide things into infinitely small segments, and any physicist can tell you that is impossible.

Very interesting things happen when things can be broken up into infinitely small pieces. For example a ball can be cut into 5 pieces and reassembled into two balls of the same size or one ball with larger volume. It's the banach-tarski paradox if you're interested.
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posted
I have also thought of the water particles and embryos but they are only specific models that represent the basic concepts expressed by 1+1=1 and 1+1>2. Thinking about these abstract concepts in terms of specific models is like thinking about axiomatic Number theory in terms of apples and oranges. It has a meaning but it misses the point.

quote:I'm not sure what is gained by making vague statements in English and "translating" them into the language of mathematics in an imprecise way.

I am not sure either.

Frankly, I can't come up with a precise formal system that would reflect the behavior of clouds and families. All I have been able to come up with in the years since I started thinking about it is specific models that do not have the aesthetic beauty that I enjoy so much when I think of Number theory, Euclidian geometry or Spherical geometry.

A spherical triangle with one corner at the north pole, a second corner on the equator at zero longitude and a third corner on the equator at 90 degrees longitude, has three right angles. People have known intuitively that the sum of the angles in that triangle is 270 degrees well before spherical geometry was defined formally with axioms. The seeming "contradiction" with the Euclidian theorem that the sum of the angles in a triangle must be 180 degrees never bothered them as long as they kept track of the context. But it was much harder to express concepts, prove theorems and make useful calculations with spherical trigonometry using an imprecise language like English.

I know intuitively that the expressions 1+1=1 and 1+1>2 have a meaning in contexts that are different from number theory. The alternate formal logic system that would allow us to express that meaning in a precise way using a small number of axioms may not have been invented yet, but I like to believe that it is possible for humans to invent such a system.
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posted
You're exploring facts, and really just countering them with more facts.

We can talk about Truth, ultimate Truth

quote: 37 Therefore Pilate said to Him, “So You are a king?” Jesus answered, “You say correctly that I am a king. For this I have been born, and for this I have come into the world, to testify to the truth. Everyone who is of the truth hears My voice.” 38 Pilate said to Him, “What is truth?”

Which has a certain philosophic flexibility.

We can also talk about facts. Now you can try to refuse to admit something occured or exists despite tangible evidence in a sort of "you have your reality, I have mine" manner. You waon't get far with that in a court of law, but you can get away with it in the court of public opinion with some people. Or you can argue that facts are concrete and have evidence. This is the way we usually think.

The problem is when people try to play both sides. A politician is on record saying something embarassing. He can deny he said it. Even when it's on the record some people will support him. Those same people would call a politician from the other side a liar for saying something that appears false and even when the evidence supports the politician from the other side will continue to call this person a liar. (And when confronted will play the "what evidence? huh? ....Well what about this other thing he said?")

Its not that facts don't exist or that Truth is relative, it's that we want them to be that way.
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posted
To the best of my recollection, I do not remember having such a conversation (insert into countless government interrogations, including interviews with Cheney regarding the Plame affair)

We have conclusive evidence that Saddam has WMD (as read in innumerable media accounts)

Your assignment, sophomore class, is to epistemologically categorize these two statements.

I will be in the teacher's lounge snorting brandy in my coffee and trying to get a peek up Ms. Boniface's legs.

posted
Viking: According to at least two major religions Jesus was a prophet. Can you elaborate on how you think your quote relates to my original post?

KL: Was Cheney quoting Clinton when he said "I do not remember" or did they both have the same teacher in sophomore epistemology?

In a rational discussion, we should allow for more than just two possibilities and recognize that the word "or" is not necessarily exclusive.
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quote: I am human and I do not think I am either a prophet or a fool. So I can not be sure.

I believe that it is impossible (as an axiom) but I have no proof that the axiom is true.

Fair enough. That as an axiom is strange in that you can prove nothing from it. So the system is the axiom. That has a nice aesthetic in a way but lacks any practicality.

quote: I know intuitively that the expressions 1+1=1 and 1+1>2 have a meaning in contexts that are different from number theory. The alternate formal logic system that would allow us to express that meaning in a precise way using a small number of axioms may not have been invented yet, but I like to believe that it is possible for humans to invent such a system.

The statements are probably better left in philosophical terms rather than mathematical terms since there is no precise translation from one to the other. It detracts from your point to dress it in the language of mathematics without a precise formal logic and definitions.

Philosophically however I think the concepts can be captured by language like the whole is greater than the sum of the parts. Language has a flexibility that allows these statements to engage the mind without a contradiction.
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quote:That as an axiom is strange in that you can prove nothing from it.

It may feel strange, but it is an inevitable consequence of Godel's incompleteness theorem that statements of this type (or it's negation) can be added as an axiom to any formal system that is powerful enough to be interesting.

Axioms and rules of inference can not be proved! They are taken for granted because people are willing to believe in them. Willingness to believe is not a rational process, the rational process begins only after the axioms and the rules of inference are accepted.

In dealing with a statement like: "it is possible for a human to know the objective truth". I see three mutually exclusive choices (please let me know if I missed something):

Accept that the statement is neither true nor false.

Add the axiom that the statement is true.

Add the axiom that the statement is false.

Trying to prove that the statement is either true or false runs into epistemological problems due to Godel's incompleteness theorem so I did not include it as a fourth option.

In a context where someone argues that they have some knowledge of an "OBJECTIVE TRUTH" that I do not accept, I prefer to add that premise as an explicit axiom just to make it clear that their argument is not rational. So if/when I am forced to choose between the following two axioms:

it is possible for a human to know the objective truth.

it is impossible for a human to know the objective truth.

I prefer to choose the later because I find it much more practical that the alternative (especially in the middle of an online debate).

In most other contexts, I am perfectly happy accepting that neither of these statements can be proved.
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quote: Viking: According to at least two major religions Jesus was a prophet. Can you elaborate on how you think your quote relates to my original post?

posted
Jesus is not the only one quoted there. Jesus seems to believe in an objective truth. Pilate seems not to.
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quote:Originally posted by seagull: In a context where someone argues that they have some knowledge of an "OBJECTIVE TRUTH" that I do not accept, I prefer to add that premise as an explicit axiom just to make it clear that their argument is not rational. So if/when I am forced to choose between the following two axioms:

it is possible for a human to know the objective truth.

it is impossible for a human to know the objective truth.

I prefer to choose the later because I find it much more practical that the alternative (especially in the middle of an online debate).

At this point we should probably define what we mean by an "objective truth". If we go by the definition that KL posted earlier:

quote:In philosophy, an objective fact means a truth that remains true everywhere, independently of human thought or feelings. For instance, it is true always and everywhere that 'in base 10, 2 plus 2 equals 4'. A subjective fact is one that is only true under certain conditions, at certain times, in certain places, or for certain people.

Then objective truths clearly exist but aren't very interesting for people who don't accept the first principles from which the truths are derived from. I'm not sure if it's possible to come up with a definition of objective truth that is independent of first principles.

One of the main problems I encounter when debating people over objective morality is that while many people have a clear source for their morality (ex: God's word, utilitarianism, etc.) they act as if their moral claims reach beyond this starting source. For example, even if I don't accept that God's word defines morality then I am still somehow "objectively" wrong when my moral beliefs differ from God's word. The only way that this could be the case is if their moral beliefs were derivable from my starting premises (or, trivially, if you redefine what "objective" means). Essentially, people want the claim "that's morally wrong" to have argumentative force for all people regardless of first principles. It's not clear to me how that's possible.
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quote:it is true always and everywhere that in base 10, 2 plus 2 equals 4

As I explained above, I used to agree with this statement before I was convinced that "in base 10, 2 plus 2 equals 4" is not "true always and everywhere".

For example, in the context of a formal system for describing population growth in a world where men and women are equally likely to be born (p=q=0.5) and each woman has exactly four children (r=4) the expression 2+2=4 would be false because starting with two men and two women the next generation would have 8 children. Note that all of these calculations are using base 10 but I chose to use axioms that assign a different meaning to the operator "plus" that makes sense in a specific context.

Saying "That's a wiki" as if that made it more believable was a really good joke.

Threads, Your last paragraph is a terrific example that the concept of "objective truth" can be used to confuse a discussion.

I believe in God as an axiom. My personal belief that "objective truth" exists is also an axiom.

By saying that these are axioms, I am explicitly stating that these are things that I believe but can not prove. I will not expect you to agree with me on "facts" that I derive from axioms that you do not believe in.

I expect the same courtesy from others who want to engage in a rational discussion.

quote: As I explained above, I used to agree with this statement before I was convinced that "in base 10, 2 plus 2 equals 4" is not "true always and everywhere".

It is true, always and everywhere, once we've defined what the terms mean, though.

As kenmeer talked about early in the thread, objectively true is more about whether or not a particular statement is valid, not whether or not the statement is true in all circumstances.

When discussing a particular event, there is an objective truth to that event. Regardless of the observer who is describing the event, the event happened in the same way. The observers might report it differently, though, depending on the view points of the observers.
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quote:there is an objective truth to that event. Regardless of the observer who is describing the event

That is so two centuries ago.

If you accept the premise of theories of relativity, observation is the only way to access "the objective truth". There are objective truths that we will never be able to access and even the truths that are theoretically accessible to a hypothetical omniscient observer can never be known by a finite observer.

Still, AFAIK, the concept of an "objective truth to that event" still means something in a relativistic world. Then you get to Quantum Theory and even that breaks down: What is the objective truth about Schroedinger's cat? Is it alive or dead? Is it even meaningful to ask that question?

quote:always and everywhere, once we've defined what the terms mean

I see an incosistency between "always and everywhere" and the use of the word once to qualify it. Would you care to clarify how you interpret the terms "always", "everywhere" and "once" without leading to a contradiction?

quote:Still, AFAIK, the concept of an "objective truth to that event" still means something in a relativistic world. Then you get to Quantum Theory and even that breaks down: What is the objective truth about Schroedinger's cat? Is it alive or dead?

Both. That's the whole point of the thought experiment.

quote:I see an incosistency between "always and everywhere" and the use of the word once to qualify it. Would you care to clarify how you interpret the terms "always", "everywhere" and "once" without leading to a contradiction?

For a given set of definitions of the terms involved, 2+2=4 is true. If you and I sit down and agree on how we are using the terms, then for every situation where the definitions were are using apply, 2+2=4 would be true.
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posted
In engineering classes, we were always told to follow the 99% rule. Essentially, if you can get within 1% of the "correct" answer it's good enough.
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quote:If you and I sit down and agree on how we are using the terms, then ...

I agree, but that is not the issue. If 2+2=4 was true always and everywhere, there would be no need for the cumbersome "If ... then ..." preamble. The fact that we have to use the word "If" implies (to me) that 2+2=4 is not true "always and everywhere".

JWatts, I took similar classes and I agree that in SOME contexts that is the right thing to do. But in other contexts ...

My work as an Engineer is used directly in designing the safety procedures for air traffic control and airport approach procedures. Do you think that if we can get 99% of the planes to land safely, that would be "good enough"?

quote:Originally posted by seagull: JWatts, I took similar classes and I agree that in SOME contexts that is the right thing to do. But in other contexts ...

My work as an Engineer is used directly in designing the safety procedures for air traffic control and airport approach procedures. Do you think that if we can get 99% of the planes to land safely, that would be "good enough"? [/QB]

I know that if you try and ensure that 100% of the planes land safely, that none will ever take off, because you will never be able to reach that magical 100% success rate.

That was kind of my point. The talk of objective truth, of 100% verifiability is just talk. Those that do must actually do.

The actual number of 9's you need for your job is just the details. Certainly 99% would be less than the current standard, however 99.999% would be 7 times greater than the current standard. And far less than 100%.

Math: 8 fatal accidents per 100 million miles Link 866 miles average distance flown Link 12.5 million miles/866 miles = 1 fatal accident per 14,400 flights (99.993% success)
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posted
Having dipped my fingers into the code that generates payroll confirmation reports, I've definitely found some of those large values of two for which 2+2=5. And when you move to multiplication, you vastly increase the margin by which a little rounding at the wrong time can skew your results.
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posted
JWatts, Thanks for the math links (the first one is really great).

Actually there is work being done on increasing safety standards (for commercial airline traffic) from 99.99999% to 99.999996%
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posted
Different proofs that all odd numbers are prime:

Math major: 1 is prime, 3 is prime, 5 is prime, 7 is prime, so by induction we can see that odd numbers are prime.

Physics major: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is a measurement error 11 is prime, 13 is prime That's good enough for me.

Humanities major: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime ...

Computer science major: 1 is prime, 3 is prime, 5 is prime, 7 is prime, 1 is prime, 3 is prime, 5 is prime, 7 is prime, 1 is prime, 3 is prime, 5 is prime, 7 is prime ...
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quote:I've definitely found some of those large values of two for which 2+2=5.

I know about roundoff. Also for a signed byte: 127+127=-2 But how exactly do you get 2+2=5?

[/QUOTE]

As I said, large values of two:

2.3+2.3=4.6 => 2+2=5

When one system adds then rounds, another rounds then adds, (or for more fund, if one does all the addition, then just rounds off what it displays) you get all kinds of interesting math.
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posted
I was talking to my 8 year old son today and he said that the temperature in the pottery oven in his class was 1800 degrees which is 18 times 100 degrees. I replied by explaining to him that multiplication for a temperature is most naturally done in Kelvin rather than Celsius or Fahrenheit.

So twice of 0 degrees celsius is 273.15 degrees Celsius and twice of 100 degrees Celsius is 473.30 degrees Celsius.

My son loves Chemistry and I don't want him to grow up believing in objectively false statements like "1800 degrees is 18 times 100 degrees".

I can't even think of a context where that statement would be both meaningful and true. But I am keeping an open mind, any suggestions would be appreciated.
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