The "Axioms" Of Nature

Methinks the stumbling block is "axiom."By definition, an "axiom" is an accepted truth that is never contradicted. In an abstraction like mathematics, we can do this without too much trouble because we get to choose those axioms by fiat.But in an observational process such as science, we can never know if what we assume to be always true actually is. We cannot observe everything. We might have it right, but we can never know for sure. We cannot simply state by sheer force of will that energy is conserved. It certainly looks like it and all of our tests seem to bear that out, but it only takes one observation to change that conclusion.It's happened before: Aristotelian physics stated that objects in motion came to rest. It certainly looked like it and all of the tests at the time seemed to bear that out, but it wasn't correct. Newtonian physics stated that objects in motion remained in motion unless acted on by an outside force. It certainly looked like it and all of the tests at the time seemed to bear that out, but even that wasn't correct. So now we have Einsteinian physics that says that motion is only relative to a frame of reference which is necessarily arbitrary and for which there is no universal. It certainly looks like it and all of the tests of the time seem to bear that out, but surely we have learned by now that we can only make that claim tentatively.In short, there are no axioms in science. There are only things that we take as axioms simply because we don't know any better. As soon as we do, we'll change them.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot writes:

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To say that there are no axioms in nature or science is simply incorrect. Things are absolutes no matter if I understand them or not.

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And as predicted, the problem seems to be the word "axiom."I am what is called a "Platonist." It is a philosophical position in mathematics that, in a simplistic sense, says that the things of mathematics are real. Take, for example, the Continuum Hypothesis. We currently do not know how to resolve it and, under our current axioms of set theory, it turns out that we can NEVER know how to resolve it.The Continuum Hypothesis has to do with the "continuum," or the set of Real numbers. To understand the "Real" numbers, we have to move back a bit.First, there are the "Integers." These are the whole numbers that we are most familiar with: 1, -12, 67, etc. Any number that has no fractional part and no imaginary part, be it positive, negative, or neither (for 0 is neither positive nor negative), is an "Integer."Next, comes the "Rationals." These are numbers that can be represented as "p/q" where p and q are both Integers. You will see that the set of Rationals includes the set of Integers because every Integer can be expressed as "p/q" where q = 1.But, there are some numbers that cannot be expressed as "p/q" such as the square root of 2 or pi. These numbers are called "Irrationals." The "Reals" is the combination of the Rationals and Irrationals. It's often called the "continuum" because it models the continuous flow of numbers across the number line.Now, how big are these sets of numbers? Leaving aside a lot of explanation, the size of the Rationals is called "aleph-null." There are a whole string of alephs, each greater than the last. Aleph-one, for example, is the next aleph and is larger than aleph-null. Since I want to keep this explanation simple, I'll simply say that there is a way to construct these numbers...they aren't just pulled out of the air. That is, just you like get "2" by adding "1" to "1" and then you get "3" by adding "1" to "2," there's a process by which you start with aleph-null and you get aleph-one. Do it again, and you get aleph-two, and so on.It turns out that the size of the Reals is larger than the size of the Rationals. Thus, it is larger than aleph-null. It also turns out that it is smaller than aleph-two. In fact, we can show that it can be no larger than aleph-one.Well, that's great, but that still doesn't tell us the size of the Reals. We know it's bigger than aleph-null but not bigger than aleph-one. Are there, perhaps, numbers between aleph-null and aleph-one? We don't know of any, but that may simply be that we're not clever enough to find them yet.Well, it was shown that if we assumed that the continuum was equal to aleph-one, then there were no contradictions to the axioms of set theory that is the foundation of mathematics. Sounds great, right?Not so fast. It was later shown that if we assumed that the continuum was not equal to aleph-one, then there are also no contradictions to the axioms of set theory.So what is the size of the continuum? A Platonist, such as myself, will tend to respond that the continuum has a size, we just don't know what it is. And while our current axioms indicate that we can never know, that doesn't change the claim that it does have a size.With this attitude, I certainly agree that there are fundamental constants in the universe. There are things that simply are true. The problem is that science, being an observational process, will never allow us to know what they are. To truly know it, we would need to have perfect observation of all possibilities which is something we can never have. We might have it right. It is possible that our observations have brought us to the actual answer, but because we arrived there via observation and our observations are never perfect, we can never honestly say that it is an "axiom" in the sense that it is something that we "know" to be always true.The best we can say is that it is something that we "think" is an axiom and are treating as an axiom, but that is only because we don't know any better.

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Now to the example. Aboard the enterprise, they were faced with a situation where they were trying to ascertain the status of other individuals aboard another ship. Mr. Spock (Rahvin) states to the captain, "Sir, there are only two logical possibilites, they are unable to respond, they are unwilling to respond". While the information was pretty much useless to the captain,it demonstrated an axiom in reality

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Incorrect. This is not an example of an axiom. It is an example of a tautology: A or ~A. A tautology is not an axiom.

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In other words no other information would shake or unsettle the axiomatic truth Mr. Spock spoke

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Again, that is not an axiomatic statement. Instead, it is a tautological statement which is not an example of an axiom.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Thanks Parasomnium.I have thought about going into teaching, but it would be for theatre, not math. Barbie was right: Math is hard. I have my degree, but it was an uphill battle.That said, it seems I spoke a bit too quickly. My basic point was right (tautologies are not axioms), but the example given isn't a true tautology as there is at least one other option besides "unwilling" and "unable":Unknowing.I may be perfectly willing and able to respond to the hail but if I am unaware that there is a hail to respond to, then it would be very unlikely of me to simply open hailing frequencies on the off chance that someone is trying to reach me.There is also the possibility that my response is going unheard. Thus, I am willing and able and, in fact, am responding but you are not hearing it for whatever reason.There are quite a few assumptions being made in the declaration that "unwilling" or "unable" are the only two options.But this goes to the basic point: The collection of possible outcomes is not an "axiom." It is simply the collection of possible outcomes and represents a tautology.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:

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If Mr. Rahain is supreme in logic then let him provide atleast a tenative solution to the only possibilites, that are not a combination of the two or a part of the two.

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You're missing the point. It isn't whether or not I can come up with a third possibility (and I actually came up with a third and a fourth). It is whether or not such a thing is an "axiom."It isn't. It is a tautology, which is something different. A tautology is logically true, not axiomatically true.Let me give an example of what an axiom would be: Euclid's Fifth Postulate.In Euclidean geometry, the Fifth Postulate states that if you have two lines crossed by a transversal such that the interior angles on one side of the transversal sum to less than two right angles, then the lines will meet on that side. Mathematicians of the 19th Century were sure that this was actually something that could be derived from the other axioms of Euclidean geometry. But it turns out that no, it can't be. It stands on its own.But here's the thing: The Fifth Postulate doesn't have to be that way. One equivalent way of phrasing the Fifth Postulate is that "parallel lines do not converge or diverge but rather maintain the same distance between perpendicular points." But, suppose that weren't true. This leads to "non-Euclidean geometry" such as elliptical geometry where parallel lines always converge and hyperbolic geometry where parallel lines always diverge.Notice that we can describe a tautology: Parallel lines either converge, diverge, or maintain the same distance between them. But that isn't an axiom. The axiom of Euclidean geometry is that parallel lines always maintain the same distance. The axiom of elliptical geometry is that parallel lines always converge. The axiom of hyperbolic geometry is that parallel lines always diverge.The tautology is logically true: The status of parallel lines is one of all the possible outcomes. But, that isn't an axiom because we logically determine the result: A v ~A.Something that is axiomatically true cannot be derived from anything: A.

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You are blinded to the fact that eloquence is not the same as an actual argument or response.

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Incorrect. I am trying very hard to be formal and thorough so as not to leave the argument to "eloquence." My statements are true not because they are phrased nicely but rather because they are precise and structured.

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You can place any term on a reality (axiom) that you wish, such as tautology you wish, but it will not change the force of its application.

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You're missing the point. A tautology is not an axiom. A tautology is true, but it is a derived truth: A v ~A. There is no other way it could be.An axiom, on the other hand, is not a derived truth but simply is. There may be other ways that it could be if we deal with other systems. The axioms of Euclidean geometry are not the same as the axioms of elliptical geometry. In fact, the axioms directly contradict each other. And yet, they are both axioms that define a specific reality.

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There is no such thing as an CURRENT AXIOM, it is either an axiom or it is not.

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Indeed, but who said that we're dealing with the reality defined by one set of axioms? Remember, I am a Platonist: The things of mathematics are real. Euclidean geometry is real. The axioms of Euclidean geometry define the way flat geometry works. But, those axioms fail to describe the geometry of an elliptical surface and they also fail to describe the geometry of a hyperbolic surface. For those surfaces, and they are just as real as flat ones, you need a different set of axioms.The reason I call them "current" is because they are the ones we use for set theory. They haven't always been that way and there are others that can be used that describe different spaces.

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If something is demonstrated to be otherwise it was not an axiom in the first place.

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You're missing the point: Axioms describe things. Different axioms describe different things. There exist flat surfaces and there are axioms that describe them. There exist elliptical surfaces and there are different axioms that describe them.Do you deny the existence of flat and elliptical surfaces?

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Axioms WILL NOT and can NEVER be demonstrated to be otherwise, as is indicated by the ones I have set out.

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Indeed. That's the point behind them being axioms. But axioms describe spaces. If you leave the space, you get different axioms.Do you deny the existence of flat and elliptical surfaces?

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All you have to do Rahain is quit exponding on the exuberance or your verbosity and give an alternate solution, even if it is theoretical.

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I did. I even gave you two: They could be willing and able but unaware and they could be responding (and thus willing and able) but be unheard.But whether or not I came up with alternatives is irrelevant. What you are presenting is a tautology, not an axiom. The reason why we have different words to describe them is because they are different things. Tautology is a logically derived statement of truth: A v ~A. An axiom is a declaration of truth: A.

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You really should pay closer attention to the the post where I said that UNABLE AND UNWILLING apply to reality and both parties.

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Fine. That still allows for at least unknowing. I can easily be willing and able to respond but if I do not know to do so, then I will not. Just because you sent a message does not mean it was received.At any rate, this is all beside the point. It doesn't matter if there are two alternatives or nine. The collection of all possible outcomes is not an axiom. It is a tautology. A tautology is logically true, not axiomatically true. A tautology is logically dervied, not axiomatically declared.

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It is clear from the situation and the reality of the situation that they were AWARE that they were trying to respond.

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Why?Actually, don't answer that. It is irrelevant. I don't care how many possibilities there are. To show you how much I don't care, I'll happily accede your claim that there are only two options.What you are presenting is a tautology, not an axiom. Tautologies are necessarily true, but they are logically derived. Axioms cannot be logically derived. That's the point. People tried to show that the Fifth Postulate was derivable from the others but finally realized that it could not be: It is an axiom. By changing the axiom, you change the space described. That's why we have flat, elliptical, and hyperbolic geometries.Since I think you seem to have forgotten: I agree with you that there are things that are simply true. I am a Platonist. There is an actual way the universe works.The only question is whether or not we can be capable of knowing what that way is. Because science is an observational process, I say that we cannot. The best we can say is that we have an accurate model that is consistent with all of the observations that we currently have. We cannot have perfect observation and thus we cannot truly know that what we think is "axiomatically" true really is.Remember my description of the Continuum Hypothesis? My position is that the continuum does have a size.We just don't know what it is.I say the universe has a way of working that is consistent.We just don't KNOW what it is...we have what appears to be an accurate model (at least for certain things), but accuracy is not identity. The best we can ever have, because science is an observational process and we can never have perfect observations, is the claim that our model is accurate. It may very well be that we have it 100% right......but we can never KNOW that. I say there are axioms to the universe. We will just never know what they are.

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You just dont know how much fun it it is to watch secular fundamentalist and the so-called scientific types that believe all things are subjective, respond to the reality of axiomatic truths. Are we having fun yet or what?

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In watching you try to psychoanalyze me over the internet? Oh, yes. I'm always fascinated when people try. I always learn such interesting things about myself. Now I'm a "secular fundamentalist." This is in contrast to me being Jewish, Muslim, Hindu, Catholic, and pagan that people have insisted I am over the years. I take great pains to keep my personal religious attitudes out of the discussion since I don't want people responding to me based on their assumptions of what I would say.I have an opinion on religious issues.You just don't know what they are.

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More rehtoric with no solutions or possible solutions.

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That's because they are irrelevant. The classic description of a tautology is A v ~A. You will notice that "A" is not defined because it is irrelevant what it is. The point is that it is a derived conclusion. But axioms are not derived. They are declared. An axiom is true simply because: A. A tautology, on the other hand, is true because of logic: A v ~A.

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This is your task my simple friend.

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Incorrect. I do not care how many options your scenario has. It is irrelevant. The collection of all possible outcomes is not an axiom.It is a tautology.

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If there are other possible outcomes then simply present them and quit suggesting that there are.

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How does the number of outcomes change the fact that the collection of all possible outcomes is a tautology and that tautologies are not axioms?It will help if you answer that since question:Is a tautology an axiom?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:

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Yes Bertot responds to you, did you think you were above it?

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Um, considering that you have misspelled my name continuously (it's "Rrhain," with two r's, not "Rahain") and considering that you responded to a whole bunch of posts in a single response, there's more than my simple desire to identify up front who I am responding to.You still seem to be confused about what a tautology is and what an axiom is. You also seem to be trying to repeat my argument to you as your argument to me. A tautology is not an axiom, right? You agree with that, right?That was my argument to you. So if you agree that a tautology is not an axiom, then why are you belaboring the point.The problem is that despite your direct statement that a tautology is not an axiom, you immediately contradict yourself by presenting a tautology and then claiming it is an axiom: That's a tautology, not an axiom: A v ~A. That's a tautology, not an axiom: A v ~A. That's a tautology, not an axiom: A v ~A.You say that tautologies are not axioms but every single example you gave of an axiom has been a tautology.Do you know what a tautology is?

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A "needless repetition of a word, phrase or idea" describes a tautology.

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Incorrect. A tautology is the collection of all possible outcomes: A v ~A.You are confusing rhetorical tautology for logical tautology. Considering that you were berating me for "eloquence," I find it interesting that you are putting forth a rhetorical definition rather than a logical one. You were mocking me as "supreme in logic" and then seem to have immediately left the realm of logic for the realm of rhetoric.

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A tautology may involve the principle of an axiom

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Incorrect. A tautology is independent of the specifics. "A v ~A" does not specify what A is because it is irrelevant. Tautology is a consequence of logic, not axioms.

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Example, you will either win or lose, thoise are the only choices, not whether you have agreater or lesser chance than someone else.

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That's a tautology, not an axiom: A v ~A.

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Actually a tautology has more to do with a persons USAGE of a truth, reality or axiom, than it does with the actual axiom itself.

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Yes. A tautology is about logic, not the specifics. That's why the classic description does not give any specifics: A v ~A. It doesn't matter what A is.

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Your usage of them in this context deos nothing to unsettle the axioms we are discribing.

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You haven't described a single axiom. You have only given tautologies: A v ~A.

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After all of this rehretoric ICANT and myself are still waiting for another solution or alternative to the two examples of axioms that have been presented.

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You haven't described a single axiom. You have only given tautologies: A v ~A.Tautologies are not axioms.

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It appears now that you have retreated to the argument that "it just doesnt matter"

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Incorrect. What doesn't matter is how many options there are in your tautology. The only thing that matters is that you have done nothing but present tautologies, not axioms.Where is your axiom?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

1.61803 writes:

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Athiest will say...

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Then athiest will say...

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Then atheist will say...

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Then athiest will say...

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Then the athiest will say...

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Who is this "atheist" you are presenting? I don't know any atheists who say anything like what you say since, assuming the atheist knows something about Big Bang cosmology, the Big Bang was neither a "quantum fluctuation" nor "from nothing."The Big Bang describes the expansion of the universe, not the creation of it.Are you simply creating a strawman?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:

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by DEFINITON a tautology is a human expression or idea applied to a reality.

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Incorrect. A tautology, by definition, is the collection of all possible outcomes of a truth statement such that all are true. The classic example is "A v ~A," but you can have tautologies with multiple propositions and which involve more than mere assertion but also include implication.It is not an axiom.

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by definiton, the word tautology would include the principle of an axiom

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Incorrect. A tautology is not an axiom nor is it related to an axiom. It is a logical derivative.

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An axion is a self-evident truth that requires no proof and is free from contradiction

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Not quite. "Self-evident" is not required for an axiom. All an axiom is is a declaration of truth and cannot be derived from other axioms.

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If that is not the correct use of the word tautology, there can be no rational application of it.

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Incorrect. Mathematicians do it all the time. Why do you think the word "tautology" is a logical term that is defined via logic?

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It is not possible for one to leave the realm of logic if they are defining, describing and applying a word by its given definition.

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When your definition is wrong, it is quite easy to leave the realm of logic.

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While it is true that a tautology is a consequence of logic, you are assuming that this is the only thing the word can have application or menaing to.

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Incorrect. I am assuming that we are discussing a single subject: Axioms. Axioms are a part of logic. Therefore, when discussing tautologies in a larger discussion about axioms, this means we are referring to the logical definition.If you wish to change the subject, that's fine, but do be aware that this will mean we are no longer talking about axioms which is the topic of this thread.

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but they are also independent of the axioms truths and realities at the same time.

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You just contradicted yourself: A tautology cannot both "include" the axiom and yet be "independent" of it. A ^ ~A is false.

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To say that a tautology is the consequence of logic and not axioms is both silly and nonsiensical, given its definition.

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Your definition is incorrect. Thus, you come to an incorrect conclusion. Given that you contradict your own argument, this is not surprising.The reason I claim that a tautology is a consequence of logic is because it is a term that is defined via logic: The collection of all possible outcomes of a truth statement such that all are true. The simplest version is "A v ~A," but any combination of propositions that are combined such that the final result of the statement is true regardless of the truth value of the propositions is, by definition, a tautology. If you want a more "eloquent" definition, a statement whose truth table contains only "true" is a tautology.Here is a more complex tautology:A ^ B = ~(~A v ~B)Let's construct the truth table:

A	B	A ^ B	~A	~B	~A v ~B	~(~A v ~B)	A ^ B = ~(~A v ~B)
T	T	T	F	F	F	T	T
T	F	F	F	T	T	F	T
F	T	F	T	F	T	F	T
F	F	F	T	T	T	F	T

Note the final column: All entries are true. Therefore, the statement "A ^ B = ~(~A v ~B)" is a tautology. No matter what the truth value of the statements A or B, the sentence is true. That is the definition of a tautology.

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The principle is reality and the demonstratable principles of the axiomatic truth applied to physical properties.

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This sentence no verb.Tautologies are not connected to axioms. A tautology is true because you can logically prove it to be so. An axiom cannot be logically proven true. It simply is.

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Hey guess what thats what makes it an axiom and not a tauology.

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Incorrect. Your statement was "A v ~A." That is a tautology, not an axiom.

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Please drag out a dictoinary and look at the definition of tautology.

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Argumentum ad dictionary? OK, if you say so: The only examples you gave were of tautologies: A v ~A. Ergo, a tautology, not an axiom. A v ~A. Ergo, a tautology, not an axiom. A v ~A. Ergo, a tautology, not an axiom. A v ~A. Ergo, a tautology, not an axiom.

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If an axiom is a self-evident truth

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An axiom is not a "self-evident" truth. An axiom is a declared truth. It cannot be derived from first principles because it is the first principle.Of the three axioms I gave, "parallel lines always converge," "parallel lines always diverge," and "parallel lines neither converge nor diverge," which one is "self-evident"?

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a tautology is a needless repetition of a word

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In rhetoric, yes, but we are not discussing rhetoric. We are discussing logic and that is not the definition of a tautology in logic. The definition of a "tautology" is a sentence whose truth table only contains "true."

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Tautologies dont describe what I ahve been describing, they simply repeat thier obvious nature.

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You realize that even by the rhetorical definition of a tautology, you just contradicted yourself, yes? If you want a "tautology" to mean a "repetition," then your admission of "repeating their obvious nature" is precisely a tautology.Which means it is not an axiom.And this is precisely what I predicted would be the problem:What is meant by "axiom"?

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I cant believe you cant see the difference.

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Given that you are trying to use rhetoric to argue logic and given that you have contradicted your own statements at least twice, I am not surprised you can't understand why I don't agree with you.Every example you have given is a tautology, not an axiom.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

ICANT responds to Straggler:

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The universe exists.

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Therefore

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The universe has always existed.

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Or

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The universe began to exist.

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That is an axiom according to the definition given by Wiki.

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Incorrect. Not only is that not an axiom, but also it isn't an axiom by the definition given by Wikipedia.Again, a tautology is a statement for which the truth table is nothing but "true." The classic example is "A v ~A."Your statement, "The universe has always existed or the universe began to exist," is of the form "A v ~A." Therefore, it is not an axiom but rather a tautology.

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What is your definition of an axiom?

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You do realize that you quoted his definition of an "axiom" in your response to Straggler, yes? What part of this are you having trouble with?You may recall my examples of the various parallel postulates. They are axioms of geometry. One is that "parallel lines never converge or diverge." This is an axiom of Euclidean geometry. "Parallel lines always converge" is an axiom of elliptical geometry. "Parallel lines always diverge" is an axiom of hyperbolic geometry.But the combination of the three, "Parallel lines either converge, diverge, or neither converge nor diverge," is not an axiom. It is a tautology.So far, nobody has given an axiom that would apply to the universe.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Agobot writes:

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I find your above statement to be too bold and would like to know what you think of:

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2+2=4

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Is this "tentative conclusion", which happens to be an axiom, ever going to change?

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Actually, that is neither an axiom nor tentative. It is a derived conclusion from the axioms of set theory.If you think you're up to it, read Russell and Whitehead's Principia Mathematica where he starts from sand and takes the 60,000+ steps required to prove that 1 + 1 = 2. [From page 362 of Principia Mathematica, Volume 1, 2nd ed. The final steps are in Volume 2: Principia Mathematica]

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true axioms do exist, otherwise the universe wouldn't be here

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I'm pretty sure most everybody here agrees with that statement.But what are they? Given the observational nature of science and given the fact that observation is never perfect, how can we ever know what they are?Edited by Admin, : Reduce image width.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Straggler responds to me:

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From a mathematicians viewpoint what is the relation between number and reality and could the axioms of number theory be considered as underpinning reality in some way?

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That's a philosophical question because the nature of science and the nature of mathematics are not quite the same. In math, you can declare things by fiat to be true, you know them to be true. You can do this because you're starting from the beginning and working forward, from causes to effects.Science, on the other hand, has to work backwards from effects to causes. As we know, this observational process can never be known to be perfect and while we can come up with what we think are axioms, they are only held tentatively because all it takes is a new observation to make us change our minds.This is where the philosophy comes in: Can we come up with a mathematics of reality? Most mathematicians (about 80%+) consider themselves "Platonists": The objects described by mathematics are real. "Number" is just as much a property of the universe as mass and energy. The fact that we use mathematics to describe the things we discover in science is, to such a philosophical attitude, testimony to the legitimacy of the claim: The world works mathematically.Not everybody agrees with this idea. My best friend, an astrophysicist, has a hard time with infinity. Oh, it's an extremely useful concept that makes the math go, but that's all it is: A concept. It isn't real. There is no such thing as "infinity" because it cannot be directly shown (to her satisfaction, at least).One possible example of evidence toward this philosophical attitude is that you can come up with multiple foundations to get to the same thing. The Peano Postulates allow you to prove 1 + 1 = 2 fairly easily. But, nobody uses the Peano Postulates for "real world" math. Instead, they use the ZFC (Zermelo-Fraenkel + Choice) axioms. Starting with them, 1 + 1 = 2 is a much more difficult proposition (Russell and Whitehead required more than 60,000 steps and over 300 pages to do it.)Note: There are only 9 axioms in ZFC. All of modern mathematics follows from those nine statements (and infinity is one of them).So to answer your questions: Platonists would most likely say that yes, the nature of reality is mathematical. Non-Platonists would not.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:

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In these definitons it should be very clear that one must start with the root meaning of the word initially to form a foundation for its use in other areas.

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Indeed, but the problem is that you are engaging in the logical error of equivocation.That is, if you have a word that has more than one meaning, it is a logical error to pretend that all meanings are equivalent and can be substituted for each other. You are attempting to do the same thing with "tautology" that creationists try to do with "theory." The word "theory" does mean "educated guess," but it also means "analysis of a set of facts." To pretend that when a scientist talks about a "theory," then he's talking about an "educated guess" is to equivocate. It isn't that the word doesn't mean that. It's that the definition of "educated guess" is inappropriate given the context.That's what's going on here. The topic of the thread is "axioms." That presumes a framework of logic for the discussion, not rhetoric. Therefore, it is inappropriate to use a rhetorical definition of "tautology." It isn't that the word doesn't mean that. It's that the context is one of logic and thus we must use the logical definition.And especially since I was the one who introduced the term into the discussion, then I get to be the one who determines what I meant when I said it. Message 75: It is clear that I am referring to the logical definition of "tautology," not the rhetorical one. I used the phrasings of logic: "A or ~A."Is there a particular reason why you think you are capable of overriding me regarding the meaning of my own words?

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Isnt it interesting when the dictionary explains in a simple sentence (Its use in logic A or not A)what Rrhain is doing so with rehtoric and much eloqunce

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Once again, you engage in equivocation. You are confusing "rhetoric" meaning "oratory" and/or "florid speech" with "rhetoric" meaning "the study of the use of language." The "rhetorical" definition of "tautology" refers to the second meaning. Thus "tautology" is akin to such terms as "anaphora," "antimetabole," "epistrophe," and "pleonasm."

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it looks like this "The candidate will win or not win"

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That is a logical tautology: A v ~A. It is not an axiom.

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In other words, even if we described an axiom as a tautology

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Why on earth would we want to do that? A tautology is not an axiom. A tautology is a derived truth. It is necessarily true, but it is only true because of other factors. An axiom is true, however, independent of all other factors. It cannot be derived from anything. If it could be, it wouldn't be an axiom.That's why we're having such a problem in this thread. You have been asked to provide an axiom of the universe. And since you seem to be forgetting: I agree that there are axioms of the universe. There is a way that the universe works that is consistent and is so "just because," without being derived from anything else.The question put to you is to please give us one of them. So far, all you have given us are tautologies, which you agreed in previous posts were not axioms.So since tautologies are not axioms (and remember that this is MY argument to you), then you have yet to actually answer the question put to you:What is an axiom of the universe?

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In a simple sentence give me an example of a tautology, no symbols, no rehtoric and no eloquence, like the dictionary definiton.

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(*blink!*)You did not just say that, did you? You quoted the definition in your own post! [emphasis added]What part of "A or not A" are you having trouble with? And since the dictionary uses it, why do you discard it as insufficient? What, specifically, is the problem with using symbols? The point is to show that the specifics of the tautology are irrelevant to the nature of tautology. Whether or not we say "win or lose" as opposed to "straight or bent" (to hearken back to another thread that involved a poster who refused to answer a simple question) is immaterial. What makes it a tautology is that it is a statement is always logically true regardless of the circumstances.It is the height of irony that you are harping upon the specific words as if they are important while at the same time inveighing against me for "eloquence." You whine about me writing well and then when I completely abandon all words and deal with the pure abstraction of logical symbology, you whine still.It would appear that what you want is not a well-written discussion nor a precise and thorough discussion but rather one where we follow the advice of Humpty Dumpty where words mean what we choose them to mean and we get to make them mean so many different things. I certainly hope you pay them extra given all the equivocation you are engaging in.Hint: That was a "rhetorical device" called a "literary reference." Let's see if you can figure out what the origin is.

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An axiom is not a "self-evident" truth. An axiom is a declared truth. It cannot be derived from first principles because it is the first principle.

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Not according to the dictionary or any observable truth or reality.

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Argumentum ad dictionary? Once again, you engage in equivocation, confusing the field of discussion with the field of logic. We're talking about "axioms of the universe" which implies a logical construction. We are asking you for what the fundamental truths of the universe are, something like "conservation of momentum." But instead, you keep coming up with tautologies: "Unwilling or unable," "win or lose," "real or not," etc. All of these are tautologies: A v ~A. None of these are axioms.

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Something declared is a human expression about reality, like the expression "self-evident".

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Incorrect. Remember, I'm the one who brought this term into the discussion. I'm the one who gets to tell you what I mean when I say it. The declaration of axioms are not "human expressions." They are simply truths for which there is no derivation. A tautology is a derived truth. That's why we don't call it an axiom. Axioms cannot be derived. If they could, they wouldn't be axioms.

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However, your play on words do not change the main force of the definiton of the axiom itself and that is that it is set against reality and physical properties, the conclusion of which are irrefutable and free of contradiction.

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Nice try. That's my argument to you. An axiom is an inherent truth. We all agree to this. Since we all agree, why are you harping on it as if that were the point of contention?No, the question is that we want you to tell us what one is. Give us an example of an axiom of the universe. "Win or lose" is not an axiom. It is a tautology. "Unwilling or unable" is not an axiom. It is a tautology. Tautologies are not axioms.What is an axiom of the universe? And how can you tell?

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These are examples of postulates in the strictest sense of the word, because they are not reality based.

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Huh? You just contradicted yourself. You claim that an axiom is "set against reality" and yet a postulate is an axiom. So which is it?

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In other words there is no physical property to pit them against, they are hypothetical, therorized contemplations.

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Are you saying that there are no parallel lines? If so, then we're going to be differing on a truly fundamental, philosophical level. I say that the objects of mathematics are real. There are parallel lines.

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In your examples there is no way to test any of the results of the conclusions other than imagination.

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Huh? "Test"? There is no "test" of an axiom. If there were, it wouldn't be an axiom. That's the entire point. That's why the mathematicians of the 19th Century were trying so hard to show that the Fifth Postulate was actually derivable from the others. They couldn't do so because it really is an axiom. It cannot be derived from the other axioms. That's why we were able to discover non-Euclidean geometry: We replaced the axiom with a different one.

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Thats not the case with physical properties and reality based axioms.

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Fine. Could you please give us an example of one and then describe how you can tell that it is an actual axiom and not just a hope of being an axiom based upon imperfect observations?"Win or lose" is not an axiom. It is a tautology."Unwilling or unable" is not an axiom. It is a tautology.Tautologies are not axioms.

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No we are discussing axioms.

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Indeed. But so far, the only thing you have given us are tautologies.Tautologies are not axioms.What is an axiom of the universe? Is the conservation of momentum an axiom of the universe? If so, how can you tell? What allows us to declare it to be true rather than conclude that it seems to be true given all observations?

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While logic is useful it can be very tenative and subjective in its conclusions, axioms are not.

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(*blink!*)You did not just say that, did you? Logic is not tentative. "Tentative" means that something is conceivably false. We may not know the reason for it being false, but there is a possibility that it might be such as having imperfect observations.Logical conclusions are not tentative. They cannot be contradicted because they are logical. That's the entire point. However, they are not axioms because they are derived. A tautology is a logical sentence whose truth table has only "true" for its entries. But to construct a truth table, you have to derive the outcome. That's why it isn't an axiom.In an axiom, you only have A and it is always true. There is no way for A to be false. That's the point behind an axiom.A tautology, on the other hand, has the option of the statements in the sentence being either true or false, but the structure of the sentence takes that into account and in the end has all outcomes being true. A v ~A. In this case, A can be true, in which case A v ~A is true. Or, A can be false, in which case A v ~A is true. Again, there is no way for the tautology to be false, but that is because it is derived from the status of A.Tautologies are not axioms. Do you agree with that statement or not?

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The sentence whose truth table contains true, is true for a reason and the reason is the axiom itself.

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Incorrect. The reason is the logic. A tautology is a derived conclusion. That means it is not an axiom. Axioms are not derivable. That's the entire point. You can prove a tautology by writing out the truth table. It is impossible to prove an axiom because that would defeat the entire purpose of an axiom: If you could prove an axiom, it wouldn't be an axiom.

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A Tautology simply reinforces the truth of it, whether it is used in a rehtorical or logical sense.

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Incorrect. A tautology has nothing to do with axioms but instead has to do with logic.

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Axioms are characterised as being truths against physical properties.

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Huh? You just contradicted yourself again.

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Hence the experssion, I exist.

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"I exist" is an axiom? Since an axiom is true by declaration, then there could be no question about it. But there is a question to whether or not you exist. In fact, the question of existence is one of the fundamental questions of philosophy.We might finally be getting somewhere. If we were to pull back to "Something exists," then we might have an axiom.

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A tautology is a repetition of this reality

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Incorrect. Again, you are equivocating. You are pretending that the rhetorical definition of "tautology" (repetition of a point) is interchangeable with the logical definition (a sentence whose truth table only has "true" for its entries).A tautology has nothing to do with axioms. It has to do with logic.

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We will wait for your simple sentence or example of a tutology to illustrate this point.

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A v ~A.That's the version that was in your own definition. Is there a reason why you won't accept the example you provided? Why is your own example not good enough for you?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Percy writes:

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it is our position that no such axioms of nature exist.

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Just to be pedantic: I claim that there are axioms of nature.It's just that we cannot know what they are due to the nature of our interaction with the universe. Since we interact with the universe via observation and since observation can never be known to be perfect, then the best we can come up with are statements that we treat as axioms but which will be tossed as soon as we come up with new observations that contradict them.F'rinstance, one could declare that the conservation of momentum is an axiom. It certainly seems to be, but that's only because we've got a ton of observations that indicate it's true. Now, from a functional perspective, this would not be surprising: Since an axiom is always true, all connections to it are necessarily true. But, that assumes we're starting from the axiom and working toward the example: Causes to effects.But we're actually working backwards from effects to causes and implications don't work both ways. If A -> B, that does not mean that B -> A. We might have an awful lot of confidence in it and might not have a single counterexample to put forward, but that isn't sufficient.I say the universe does have axioms.We just don't know what they are.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Agobot writes:

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Here is one axiom of nature:

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LIFE ENDS IN DEATH

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Now i challenge everyone and anyone of your camp to prove me wrong with an example from all earth's history that overturns this axiom.

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Oh, really? What about single-celled organisms that reproduce by fission? When a single-celled organism divides, does it "die"? If not, then it would seem that there has been an unbroken chain back to the beginning, stretching billions of years.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:

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Reality is not a THEORY

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Logical error: Equivocation.You are using "theory" in either a colloquial or a strict scientific sense. When I said "set theory," I was referring to a mathematical sense: A body of principles, theorems, or the like, belonging to one subject.

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Existence is however a PERFECT example of reality, even if we dont understand all its parts or functions.

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Nice try, but that's my argument to you. I'm the one saying that we cannot know what the axioms of reality are, even though they exist.It has been put forward that your thesis is that we can know what the axioms of nature are. If this is your thesis, it would be nice if you could give us one. If this is not your thesis, please let us know.If we cannot know what the axioms of nature are, then how can we make reliable conclusions about the world around us?

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The axiom is the reality of existence

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"Something exists"? That's your axiom? Well, I think I could go along with that one. There actually is some use to it (since it is the foundation for what we experience). But that doesn't tell us if the world we experience is what actually exists or is simply a simulation and that what really exists is something else.This is known as "Cartesian Doubt," so named because René Descartes covered the topic in his Pensées. Descartes eventually does away with the idea that we are, as he describes it, "plagued by demons" by stating that if the simulation is perfect, if it is impossible to ever transcend it and see the cause of the simulation, then there is no difference between the simulation and "reality." Since a difference that makes no difference is no difference, then the simpler answer is that reality is real.But, back to the direct question: Is that an axiom of the universe? "Something exists"?

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The only way anyone could ignore the force of the conclusions that flow from this reality is to imagine that things dont really exist at all

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And why is this unacceptable? Are you familiar with Plato's Parable of the Cave?

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which only removes it form one place to another.

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Indeed, but what does that say about our reality? If all of this is a simulation, then what is the nature of reality? And if all of this is a simulation, what is to keep whatever it is that is running the simulation from changing it? And if it can be changed, what is to let us know that it has been? If everything is a simulation, it is conceivable that it was initiated two seconds ago with everything we think of as history being pre-loaded into the system.How can we possibly make any sort of accurate statement about reality then?

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To do this however one would need to avoid or discard all the evidence of existence in the first place.

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You say that as if doing so would be a problem.Why?

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.

Bertot responds to me:Well, no...no, he doesn't. There's just a lot of equivocation and avoiding of the subject. So let's start over:A tautology is not an axiom."Unwilling or unable" is a tautology. Ergo, it is not an axiom.Can you give us an axiom of the universe? I say we cannot know what they are. Your thesis seems to be that we do. So help us out:Give us an axiom of the universe.

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Rrhain

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Thank you for your submission to Science. Your paper was reviewed by a jury of seventh graders so that they could look for balance and to allow them to make up their own minds. We are sorry to say that they found your paper "bogus," specifically describing the section on the laboratory work "boring." We regret that we will be unable to publish your work at this time.