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Are axioms subjective?

  1. Nov 10, 2009 #1

    apeiron

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    Are the axioms of math subjective? If they are, then logically all the formal consequences that flow from them are also subjective. Thus there can be no objective mathematical facts.

    Someone said this:
    Someone then replied this:
    But then later made the contrasting statement:
    A priori knowledge is knowledge claimed to be independent of experience. So it would be "internal" in being derived by reason and yet also (it is argued) objective - ontically true rather than merely an outcome of human modelling, human construction.

    Of course, subjective~objective is only one way of framing this dichotomistic distinction. Others (which could be better) include immanent~transcendent. Or in more recent epistemological debate, internalist~externalist.

    And broadening the terminology further, these issues have been discussed in terms of the analytic~synthetic and the contingent~necessary. Or the Platonic dichotomy of chora~form. And of course, the dichotomy epistemology~ontology breaks across the same lines.

    The point is, subjective vs objective is not a hard and fast distinction here. But there is a broad understanding of what the distinction involves.

    And so the question is: are axioms subjective (as I would argue is "proven" by Godellian incompleteness)?

    And if so, then all maths is constructed, even if we may feel the consequences of axioms have an "objective" or a priori truth - self-evidently true in the light of what has been assumed to be true?
     
    Last edited: Nov 10, 2009
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  3. Nov 10, 2009 #2
    Of course they are subjective (I would rather use the term optional though, as in: you may accept them). If you deny the existence of an infinite set, then it would be rubbish to say that the set of natural numbers exists. If you claim that two sets can be different despite having exactly the same elements, then the set-builder notation becomes invalid. When people talk about mathematical facts they usually leave out massive amounts of information because this information is implied or trivial for the other person to work out himself (definitions, trivial steps in proofs, analogous parts of proofs, axiom systems, etc.).
    One of these things we leave out is our axiom system. If a theorem reads "P is true", then what we normally mean is "Assuming the axioms of ZFC, P is true", but we leave this out since all mathematicians agree that unless otherwise noted that is the axiom system which we use (in some fields such as set theory we may get a bit more explicit and often only assume ZF, or even other systems). Math doesn't say "a product of compact spaces is compact" (Tychonoff's theorem), it says "if you agree with our axiom system ZFC, then you must also agree that a product of compact spaces is compact".

    This may sound a bit purist, but in math we don't care about the "real world" except possibly for motivation. We care about our internal constructed world. Of course there are things we can't tell about this world (as per Godel's incompleteness theorems for instance), but if I from my axioms can deduce 1+1=4, then that is acceptable. If I can also deduce 1+1=2, then we have a contradiction and that is surely not acceptable unless of course 2=4 which would also be fine, but I don't care whether the natural numbers works for counting apples or not (personal experience tells us though for small natural numbers they do seem to work for this purpose though, but this is more a concern for physicists or engineers perhaps).

    I don't suppose ZFC in the real world because there is no such thing as a "set" in the real world, but I like to consider a logical system in which they are true and then see where that leads me.

    If someone spots a similarity between a mathematical system and a physical system, then that's fine, but if the similarity breaks down at some point that doesn't mean that the mathematical system or physical system is broken, but that the similarity wasn't complete. For instance people believe that if we have x apples and combine them with y apples, then we get x+y apples, but this makes no sense when x=10^1000 y =5 because there aren't 10^1000 apples in the universe or the possibility of making 10^1000 apples from the few molecules we have. Even for rather small numbers such as x=y=10^40 I suspect gravity would turn them into a planet-like object giving us x+y>z where z is the resulting number of objects in the physical system.
     
  4. Nov 10, 2009 #3

    apeiron

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    Rasmhop, I would agree that it seems obvious that axioms are subjective. And that the Godellian failure to resolve paradoxes by "internal means" proves that. So in practice, even the majority of mathematicians would take a pragmatic view of axioms. They only say "If this applies, then this follows".

    And this would then lead us on to the even more current question of how our axioms are best developed?

    What is the best way to do it so that our axioms are "as true as possible"? Or at least, if we really stick to positivism, "as useful as possible"?

    But first, let's hear if others can supply good arguments for why axioms are not subjectively and immanently emergent, but instead objectively and transcendently true.
     
  5. Nov 10, 2009 #4
    What do people mean by "mind-independent" and "objective"? One example would suffice.
     
  6. Nov 10, 2009 #5

    apeiron

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    I think this is actually deceptively tricky.

    The naive view would just be to insist the world exists in independent fashion - I refute (solopcism/idealism) thus, said Dr Johnson, painfully kicking a large stone. It seems straightforward commonsense that the objective is independently real and the subjective is the dependent illusion.

    But QM clearly entangles observers with the observed. And modern approaches to epistemology have had to take account of that physical fact (or "fact"). Hence the efforts of von Neumann, Pattee and others on the issue of the epistemic cut, the modelling relation, etc.

    Yet, again, this becomes a level of issue only after we establish a general agreement about the idea that axioms are subjective and so the consequences of axioms remain subjective.

    Then, in reflexive fashion, what we learn about "fundamental reality" as a result of those axioms can feed back into the whole epistemic process. So Newtonian physics was founded on a set of axioms, which have since been challenged or modified. Now we are in a position to ask how new knowledge changes things.

    Of course, there is other new knowledge that needs to be incorporated into epistemology (which did not stop with Descartes and Kant). We have cognitive neuroscience. We have systems science and complexity theory.

    We are really asking what takes place in the mind of a mathematician when s/he understands something apparently fundamental? What is the process involved?

    If it is just choices about axioms and explorations of their consequences, then that justifies a careful examination of the choice-making procedures. A meta-epistemology.

    But if axioms are in fact objective truths - well, I'm not exactly sure of the consequences of that for epistemology, never having taken the possibility really seriously.
     
  7. Nov 10, 2009 #6
    I see what people mean(or more aptly believe) by 'objective', but i don't see how that can be mind-independent. This belief/assumption has got to be a time-related extrapolation involving past events, dinosaur bones, etc. Anyway, my brain is also telling me that its real and objective. My brain is also teling me that the brain is the most important and "I"(whatever that is) see no way how i could be able to bypass my brain(the information processor) and ascertain what mind-independent reality might be and if it exists.
    It's not just idealism vs materialism, there are nuances in-between that are not captured in this dichotomy. In fact, both positions can(and seem to be) be correct at the same time and describe the same phenomenon seen through different angles.
     
  8. Nov 10, 2009 #7
    I never look in the philosophy forum, but this was the last thing posted under when I was looking at the board.
    Anyway, what exactly are you guys talking about? How can having one apple and one orange not give two pieces of fruit? That's objective, right? Unless you are arguing that numbers don't represent what they do and that's just too weird.
     
  9. Nov 10, 2009 #8
    Actually just the other day I took an apple and an orange, and suddenly I had a pear... Well no, but who is to say that it won't happen? All our previous experience tells us that when we take an apple and an orange, then we have two fruits, but what if the probability of getting two fruits is simply 99.99999999% and you may one day find two fruits that combine to form one? My mom has played the lottery as many times as you have combined apples and oranges I guess, but she hasn't won yet. Can we conclude she never will? (no just that it's unlikely based on prior experience) Have you ever combined 925 apples with 1075 oranges and manually confirmed that you get 2000 fruits? If not then how do you know that your previous experience generalizes to this case. The reason we "know" this works is that we say that when dealing with "sets" of fruits we presume they satisfy the axioms of ZFC and we presume counting objects satisfy the Peano axioms where the successor function's interpretation is adding an object. Under these assumptions math can help us conclude that 1 + 1 fruit combine to form 2 fruits, but if it turns out that there exists two bags A and B of fruits such that if we add one object to both A and B, then the new bags are equally large, but the original bags weren't, then we have violated one of our Peano axioms which means that our assumption was wrong. Note however that these assumptions are yours to make and not something automatically true. I may believe that counting will stop to work like numbers in year 2030 and there's nothing you can do to prove my subjective belief wrong until we reach that time.

    I do not argue that numbers do not represent what they do, because that's a tautological statement, but I argue that we can't be sure whether they represent counting objects unless we define counting objects in terms of numbers and not in terms of physical objects. For instance most experimental data suggests that there are a finite number of particles in the universe so any number greater than that could not represent the process of counting anything physical. All we know is that in most cases we have observed counting objects and numbers behave similarly, but all our experiences are with small numbers of objects and to be honest most humans have few (<10^15) experiences of counting objects and verifying the result so there may be a minute chance of counting violating basic arithmetic some day.
     
  10. Nov 11, 2009 #9

    qsa

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    I think this problem has already been beaten to death, started in 1850, now it is just one more game.mathematicians and philosophers don't doubt the truths of numbers and their relations,which imply their objectivness. They only try to make it on a more rigreous ground so that they satisfy themselves as to the consistency. And to see exactly where these truths come from just like we try to find why we see apples fall. the fact they fall is considered by almost all to be true and objective, but why ,that is another question.
     
  11. Nov 11, 2009 #10

    apeiron

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    I would disagree. Yes, there is a way of thinking about reality that is highly successful. Yet it arose out of a critical fork in the path. And the path we didn't take now seems like the one we need to explore too.

    As far as number goes, for example, there is the fork in the path where you either take number as simply existing, or where you treat number instead in terms of limits to processes.

    The choice made was pretty clear with Cantor and infiinity. Infinity used to be a limit to a process of counting. Then Cantor said, you know hey what guys, let's just treat infinity as an existent object and see what the mathematical consequences would be. Nowadays, only a fool would try to argue any different.

    Well, actually, there are a few of us who are interested in taking that other path and seeing where it can lead.

    A view of reality based on the idea of asymptotic limits is quite different from one based on that of existent objects.

    And if axioms are subjective, then we have to admit choices do get made. It is a valid project to retrace our steps and explore other paths more fully.

    But this can't be done unless you are fully aware of the choices that got made. All the mathematicians and philosophers who don't doubt the objectivity of their current choice (must be a different bunch from the ones I associate with) are acting deluded.

    The way they model reality works great in many ways, and so they should be free to continue developing this approach. But the failures of this approach are also well-known - the self-referential paradoxes, the problems with chance~necessity, the flip-flopping between monadism and duality, the problems with initial conditions....
     
  12. Nov 11, 2009 #11

    qsa

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    axiom are assumed to be true and objective by definition. but sometimes they fail in trying to answer certain questions. all that means is that the axioms are not complete.
    saying that they are subjective, implying they are not true and hence cannot be used means you want to start a new math, good luck.
     
  13. Nov 11, 2009 #12

    apeiron

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    That is nothing like what subjective means here. True vs not true are both objective notions. The subjective view is "seems likely vs seems unlikely".

    Or being more careful in the positivist, pragmatist, spirit of science (and pragmatists philosophers like CS Peirce), it is all about mental constructs that are useful vs ones that are not useful.

    So a belief in number has proved useful. A belief in god or psychic powers or whatever proves much less useful.

    But forget all that. Can you supply some actual argument why we should believe axioms to be objectively imposed rather than subjectively chosen? Apart from "every mathematician and philosopher clearly believes this".
     
  14. Nov 11, 2009 #13

    qsa

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    it would be foolish to just decide and choose anything that does not hold any amount of truth(objective) and expect much result to follow that will be usefull and consistant.
     
  15. Nov 11, 2009 #14
    Ya know all this axioms talk has lead me to actualy look at some of these axioms and they honestly seem a bit unreal. Unreal in the sense that they had to be put on paper in the first place. Example ?x=x? and ?x=y y=c x=c? Well no duh... Do people really not realize these are true at birth? Also I could make an argument that x does not equal x if I were to say that this *x* is not the exact same as this *x* because clearly they are not in exactly the same location and therefor can not be exactly the same. However the relevence of that argument is almost none or possibly less then none.'

    It also seems to me that math would not follow rules in its form as it is in itself the rule.
     
  16. Nov 11, 2009 #15

    apeiron

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    Well again, this would be the naive approach.

    If we accept we are modelling, then this also leads naturally to idea that the modelling has to be meaningful in some light. It has to achieve that purpose.

    So what become our choices? I would say they can be divided into a desire for truth vs a desire for control.

    So a thermostat embodies a model of the world in the curvature of a bimetallic strip. It is brilliant at achieving the purpose of control. But it knows very little of the "truth" the room it is in, the heating system it operates. We can imagine creating some more complex computer sensor, but why bother. All the extra data would confuse the poor thing, make it more likely that it could go wrong.

    And much of science/maths has the same actual purpose - to deliver models that control. To have the kind of knowledge that allows us to build machines, create technology - take the simple view that restricts the space of possibility, stops us getting too confused by the wider context of what might exist.

    Minds/brains in animals also embody the purpose of control. Minds exist to make decisions, execute actions. Not to sit and contemplate the "deep truth of reality". Animals are not philosophers.

    So arguably, this demand for objective truth is quite unnatural in fact. And it is not actually what mostly happens.

    Axioms are an example of this. They are not really about what is true but what is useful to assume. They seem special because they do try to maximise a number of key modelling qualities - generality, invariance, crispness. But they take hold in the human imagination because they are judged on their effectiveness.

    If an axiom was true but ineffective, I'm sure you would realise how few people would bother with it.

    So this is part of the claim that axioms are subjective. There is a process of choice. And also the choice brings in human-scale purposes. If kings want to control their peasants, then they need scribes who can count the annual harvest. Before there was such a purpose, who cared about number beyond one, a few, the many?
     
  17. Nov 11, 2009 #16

    apeiron

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    It is not so simple. Take three apples. X may equal Y in size and ripeness close enough, but C has been dropped and bruised. I don't consider them all equal.

    But axiomatically, we can chose to jump to idealisations. We define things to be equal (make that epistemic cut with reality) and get going spinning consequences of this assumed fact.

    You could call it objective, but clearly it is a subjective stunt you just performed on reality. In reality, such perfection don't exist. Objectivity is all in the eye of the beholder.

    People here just keep claiming number is objective, axioms are objective. Let's hear some actual reasoning.
     
  18. Nov 11, 2009 #17

    qsa

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    we assume them to be facts and if no contradictions are found then we have high confidence in our assumtions.without objective truths no social or physical existense is possible. jump in a 1000 F oven and lets see if you think it is all in our head.
     
  19. Nov 11, 2009 #18
    Remember, axioms themselves are even modified over time. Before the Scientific Method became popular, deductive logic was extremely popular. Then many scientists said deductive logic was useful, but not good enough because it's always possible your starting assumptions are false, and so decided it was best to come up with ways to test things. Math axioms are pretty good, but can be modified over time.

    Deductive logic isn't in itself bad. It's just that deductive logic's conclusions is only as valid as the starting assumptions are correct. Deductive logic doesn't mean the conclusion is true, but rather if the starting assumptions are true then the conclusion is guaranteed, as opposed to inductive logic.

    Even some of our most precious axioms can be modified over time, just like Newton's Laws of Motion have found situations to be modified. The founders of Science even found many axioms to be modified. Aristotle was big into logic, and many of his beliefs were later to be found false, even if many were true. Again, deductive logic uses premises that eventually start with induction.

    The only way to prove an axiom is universal is to be everywhere at once and at all times to make sure. Otherwise, we only "fail to disprove" an axiom. So we can use deductive logic, but somewhere down the line you have to eventually use inductive logic as a starting point.
     
  20. Nov 11, 2009 #19
    If you find yourself with some kind of problem, how can you tell if it is due to the system or due to what you are trying to accomplish?
     
  21. Nov 11, 2009 #20

    apeiron

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    Totally missing the point.

    The commonsense default position would be that reality is real (objective, independent of our existence) and that our knowledge of it is subjective (constructed, dependent on reality's existence and shaped according to our position within that existence).

    But then philosophy, in an attempt to be rigorous, asked - given we plainly cannot see things directly, what can we in fact be certain about as a starting point.

    Descartes answer was, in principle, virtually nothing. We can be pretty sure there is us doing the thinking, asking the question. Everything after that has to be treated as a choice - not a free choice, some postmodern anything goes deal, but a suitably constrained choice.

    To then claim maths is also objectively certain would demand justification. Kant tried to recover other certainties, but I don't find his approach convincing. You can re-run those arguments or some others if you wish. I invite you.

    But I am running the counter-position that cartesian doubt is "fundamental" and so the correct response is to pay attention to suitably constraining the modelling choices we make.

    This is what epistemology is about.
     
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