The "Axioms" Of Nature

We would adapt as we did by developing non-Euclidean geometries that deal with other alternatives.Right now we don't have the evidence so we, for the most part, take reality as an axiom.Sure the alternatives are considered but not with a fraction of the percent of resources directed to working in a "reality" where the axiom is taken as given -- that is, as an axiom. So we aren't disagreeing for the most part? That is what I think we do too. (almost all the time).

Percy writes:

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Me, I think the problem is that axiom has more than one definition.

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With regard to the philosophy of science, no, not really. The thing is that most scientists work science without any formal training in the philosophy of science. In and of itself, this isn't too horrible: Most people can do math, even complex math, without any formal training in the ZFC axioms of set theory. When you're introduced to arithmetic in school, you are given examples of one and one equaling two, but you certainly aren't given the formal proof of it. It's way too complex and the fact that it can be proven is sufficient to start from there.That is, we treat the foundations of arithmetic as axioms even though they're not. Because they can be derived as true from the axioms of set theory is sufficient for most purposes. But treating it as an axiom is very different from it actually being an axiom. That's the point I am making (which I think Straggler agrees with): There are things we treat as axioms, but that is only because we don't know any better.

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I understand that Rrhain is using this precise difference of opinion to show Bertot why he is wrong

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Not quite. It's more that I am making the distinction between assumptions and conclusions. This comes up every now and again in our debates here with creationists claiming that we are assuming evolution when in reality we are concluding it.

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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.

NosyNed responds to me:

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And what would science do if we found evidence to the contrary?

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We would adapt as we did by developing non-Euclidean geometries that deal with other alternatives.

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Incorrect. The Fifth Postulate is an axiom. Non-Euclidean geometry doesn't change that. Instead, it completely discards it and replaces it with another axiom.In science, new observations give us new understanding about how the world works, but they don't change the observations we have already made. When we moved from Aristotelian to Newtonian to Einsteinian mechanics, none of the things we had observed before changed. You could still run the same experiments and come up with the exact same results. As Gould put it, apples didn't hover in mid-air, waiting for us to figure out how gravity worked.But if you change an axiom, reality changes. In Euclidean geometry, parallel lines never converge or diverge. But if you replace that with a the axiom from hyperbolic geometry that parallel lines always diverge, then the case of you having two lines that never converge or diverge means not only are they no longer parallel but also they are no longer straight. The entire structure of everything is altered if an axiom changes.

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Right now we don't have the evidence so we, for the most part, take reality as an axiom.

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No, the philosophy of science concludes reality. It is not taken as an axiom. Now, we can draw true conclusions from true statements that are themselves true conclusions of earlier true statements (everything eventually goes back to the axioms), but the fact that we start in the middle doesn't make those middle things axioms.That is, let A -> B. We can then engage in a new process to conclude B -> C. This doesn't make B an axiom because B is derived from A. For practical purposes, we're treating B as an axiom because it is awfully tedious to have to start from sand every single time, but B isn't really an axiom. If we have to, we can go all the way back to the beginning.

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So we aren't disagreeing for the most part?

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I don't think we are. The point is subtle, I admit, but it is important. As Straggler points out: There is a difference between starting from perfect knowledge (represented by an axiom) and starting from imperfect knowledge (something that we treat as an axiom but is really just the result of a collection of imperfect observations that so far have yet to be contradicted).I deal with this necessity of analysis all the time in my work. When I train new people, I often give them a question (that I admit I stole from a Foxtrot strip): You see, if you were given the setup and were asked to determine the average speed of the train, you'd note that 10:00 am to 2:00 pm is 4 hours, divide 180 miles by 4 hours, and come up with 45 mph.Ah, but is that really the answer?

• Did the train arrive on the same day?
• Are the stations in the same time zone?
• Did the train take the short route or did it go around the globe?
• If it took the short route, did it go along the curved surface of the earth or tunnel straight through?
• Is the track direct or does it meander such that the stations may be 180 miles apart but you have to go to Station C in order to get to Station B?
• Are the clocks working at the stations or does it always say "10:00 am" at Station A and always say "2:00 pm" at Station B?
• If the clocks are working, are they accurate?

The assumption of the clocks being working, accurate, stations in the same time zone, 180 miles being the exact distance the train traveled, etc. are all perfectly reasonable assumptions to make, but we should not take them for granted or forget that that is all they are: Assumptions and they may not be true. The point is that because we do not have perfect knowledge of what it is that we're dealing with, we have to be careful.Treating something as an axiom is not the same thing as it actually being 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.

It feels to me like you and Bertot are each arguing that your own way is the only way to look at things, and you're both taking the tack that if you can just convince everyone else to accept your way of looking at things then you win the debate. The significant difference between you is that your way makes sense while Bertot's is just made-up nonsense, but I think you're both wrong in thinking there's only one way to approach discussion of the issue.But to reduce confusion I think only one definition of axiom at a time should be in play in a thread. Bertot's having a field day pointing out the inconsistencies between the viewpoints expressed here.--PercyEdited by Percy, : Grammar.

Strangely enough they are axioms. In line with what Percy said there are several different meanings of the word axiom. In mathematics an axiom is simply a definition. It explains the nature of the mathematical object you are talking about.One can formulate arithmetic independent of formal set theory with Peano's axioms. These axioms describe an object called the naturals obeying certain conditions, elements of which are called numbers. It can easily be proven that these numbers produce all of the results of school mathematics.However if one takes Zermelo-Fraenkel-Choice set theory, then these axioms describe a universe of sets obeying certain conditions. It is an easy matter to prove that there is a set inside this universe which obeys Peano's axioms and hence all of school mathematics is produced by ZFC set theory.However both ZFC or Peano's axioms are axiomatic systems. Peano's axioms really are axioms. ZFC is more powerful in the sense that it contains a system which obeys Peano's axioms, as well as other systems (such as the real numbers and other mathematical objects).

Percy responds to me:

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It feels to me like you and Bertot are each arguing that your own way is the only way to look at things

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I wouldn't say that. I'm simply pointing out that people have been thinking about this very question for literally thousands of years. The philosophical traditions that we have all eventually come back to this question. The Buddhist idea that everything is naught but illusion is quite different from the Western view that there is a separate reality apart from human experience.Bertot and I both seem to be more along the lines of Western philosophical underpinings: There is a reality and it behaves in a consistent way. The question we have is whether or not we can know what that way is or can we only approximate it at best.But agreed: We need a single definition of "axiom." But since I'm the one who introduced "tautology" into the discussion, I'm the one who gets to say what it means in this context.Tautologies are not axioms.

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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.

I'm not sure why you think you get to choose which definition of axiom is in play, but you could certainly do so if you propose the successor thread.--Percy

Percy responds to me:

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But since I'm the one who introduced "tautology" into the discussion, I'm the one who gets to say what it means in this context.

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I'm not sure why you think you get to choose which definition of axiom is in play

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Um, what part of "I'm the one who introduced 'tautology' into the discussion" are you having trouble with? Here are the exact words I used when I brought the term up (Message 75): This was in response to Bertot's statement (Message 7): So I introduce the term, define it immediately after introducing it, and somehow I'm not allowed to say that I meant what I said? You might be able to say that my definition and use of the term is a non sequitur, but how do you justify contradicting me on my term used in my sentence with my definition?Now, far be it from me to resort to argumentum ad dictionary, but Bertot seems to like it and his own quotation of the dictionary definition of "tautology" is "A or ~A," which is exactly what I said.So again, how is it that I am not the final authority on what I meant when I said what I did?

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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.

You are, but you can't force your definition of axiom on everyone else, that is unless you propose the topic yourself and make it one of the preconditions, hint, hint.--Percy

Well I would be interested in a follow up thread that explores the "axioms of scientific investigation".In this context by axioms I mean the unprovable assumptions that are required in order for scientific empirical investigation to be considered both valid and worthwhile.For example the assumption that an objective reality that we all indvidually and subjectively perceive does actually exist.But I am not sure how much appetite there is for yet more discussion of axioms........?

Well, if you give me some time, I'll see if I can dig up my old philosophy of knowledge/science texts and see if I can come up with a good original post regarding what philosophers have said about how we can know what we know.

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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.

I can completely understand if everyone is talked out on the subject, but it would still be nice if there was a thread where everyone was on the same page as to which definition of axiom was in play. And I like your suggestion that it should be the one about the "unprovable assumptions that are required in order for scientific empirical investigation to be considered both valid and worthwhile."It sounds like Rrhain might be volunteering, thanks Rrhain. Don't go too far over our heads, or at least mine.--Percy