Are Axioms of Math Subjective?

[apeiron]

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:

... it is hard to deny "4=2+2". There seems to be an objectivity to "2+2=4" that is independent of sense experience.

Someone then replied this:

Sure their are mathematical facts, there are all kinds of facts, but your claim was that they were objective. Objective facts are independent of mind. Mathematics is something we learn when we are very young, so its not really surprising that we take it for granted.
That's because 2+2=4 is not an axiom. Its a formulation that relies on axioms that were abstracted from exprience, and taught to you when you were young.
All kinds of things seem objectively true... because we are used to them, because we grew up with them and have developed an intuition about them. Intuition is not objective, it is, by definition, subjective.

But then later made the contrasting statement:

There are plenty of people on this board who would say that math is objective, a priori knowledge which means internal and objective.

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?

 

[rasmhop]

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.

 

[apeiron]

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.

 

[WaveJumper]

What do people mean by "mind-independent" and "objective"? One example would suffice.

 

[apeiron]

What do people mean by "mind-independent" and "objective"? One example would suffice.

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.

 

[WaveJumper]

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.

 

[theCandyman]

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.

 

[rasmhop]

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.

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.

 

[qsa]

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.

 

[apeiron]

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.

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...

 

[qsa]

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.

 

[apeiron]

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.

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".

 

[qsa]

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".

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.

 

[magpies]

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.

 

[apeiron]

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.

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 realize 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?

 

[apeiron]

Example ?x=x? and ?x=y y=c x=c? Well no duh... Do people really not realize these are true at birth?

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.

 

[qsa]

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 let's see if you think it is all in our head.

 

[27Thousand]

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 let's see if you think it is all in our head.

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.

 

[theCandyman]

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?

 

[apeiron]

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 let's see if you think it is all in our head.

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.

 

[JoeDawg]

saying that they are subjective, implying they are not true

That is not what subjective implies.

 

[Hurkyl]

I'm going to argue that math is objective. The subjectivity people have been discussing in this thread is not part of math -- instead, it's regarding the application to "real world" problems.

For example, there is no room for subjectivity in a question like "Is the statement P a theorem of first-order Euclidean geometry?" There exist explicit and completely specified algorithms which will give a (correct) yes/no answer to this question.

Subjectivity only comes into play when we start asking questions like "Does the real world obey Euclidean geometry if we interpret the words 'point', 'line', 'incident', 'between', 'congruent' appropriately?"

(I chose this theory for simplicity of exposition, because it really is complete -- this theory is too weak for Gödel's incompleteness theorem to apply)


For the sake of pedantry, subjectivity does have its place in mathematics -- things like deciding what presentations are best, what lines of research to pursue, et cetera.

 

[qsa]

The history of philosophy is well known. philosophy deals with the nature of truth not that such a thing exists. Although at times in the middle of all the arguments you get the impression that somebody is denying that any truth can exist and if they do we have no access to them ( at least with 100% certainity). Yet, no matter what philosophers have told us, we do our science with that basic assumtion and we have come a long way; even to the point of discovering the origin of reality.

the question of how the mind acquires truth is interesting but not prohabatitive, basicly because the mind is such a complicated machine. Even this problem is been chipped at everyday as our technology become more and more sophisticated. Again, some people taking the opportunity of this complicated machine to devise wild theories, like the mind is reality, reality is like the mind and ...so on. Philosophy is in this sense very usefull to clarify all possible scenarios no matter how flimsy.

 

[Pythagorean]

I tend to believe that any time we use language or communicate, there is bound to be subjectivity involved.

Anyhow, a more appropriate question might be "must axioms be objective". I don't think they are required to be.

We can whittle and pick away at the different perspectives of axioms and whether they are subjective or objective, but in the end, these are ideals, like a perfect conductor or a perfect insulator... they likely don't really exist.

Instead, things can only be compared: more subject or more objective. I think mathematics is one of the more objective things out there (not to be confused with perfectly objective).

 

[apeiron]

For example, there is no room for subjectivity in a question like "Is the statement P a theorem of first-order Euclidean geometry?" There exist explicit and completely specified algorithms which will give a (correct) yes/no answer to this question.

It has already been conceded that the consequences of axioms may have that inevitability, that loss of choice by us, which we would call objective. Which is why the question is about the axioms themselves.

So it is not what the algorithms compute, but the initial choices that led to the algorithms.

In Euclid's case, he started with “…any two points can be connected by a straight line.”

Now it is obvious how much has to be presumed in this statement, even if we feel it is a very reasonable and useful statement.

Objectively speaking, all sorts of consequences must follow if this statement is true. But the statement itself is developed subjectively - there were choices being made. And experiences of the real world were being idealised or generalised.

Now what is a point. for example? A zero-D object? We can see how the idea was derived by taking our impression of a world full of things at locations and shrinking down the notion of locatedness within a dimensionful context until we just have pure location with no extension, shape or motion. We've made a bunch of choices of subjective properties to ditch in order to create a useful mathematical fiction.

Could a different choice have been made here? Euclid said just accept a point is a location that exists without extension. But it seems at least possible for an alien race, who happen to love condensed matter physics (the Laughlinians of Laughlinia) to have instead defined point-likeness as the limits of a process of dimensional constraint, so a quasiparticle like defect in a continuous fabric. Instead of points axiomatically existing, axiomatically they become the limits of what can exist (and so precisely what does not exist).

Perhaps after going through the arguments, it will come to seem that subjective~objective just is not a good term to capture the essential distinction here. I would agree that the terms are rough rather than precise.

But for the moment, the focus has to be on the status of axioms. Can they too really be something we know to be objectively true in "a no choice about it" fashion? Even if humans did not exist, the axiomatic truths would have to actually exist? And that alien minds who arrived at much the same outcomes, but by different axiomatic routes, would be objectively wrong?

 

[apeiron]

I tend to believe that any time we use language or communicate, there is bound to be subjectivity involved.

Anyhow, a more appropriate question might be "must axioms be objective". I don't think they are required to be.

We can whittle and pick away at the different perspectives of axioms and whether they are subjective or objective, but in the end, these are ideals, like a perfect conductor or a perfect insulator... they likely don't really exist.

Instead, things can only be compared: more subject or more objective. I think mathematics is one of the more objective things out there (not to be confused with perfectly objective).

Reasonable comments. But the problem was - as supported by the quotes in the OP - that many people have been taken an unreasoned approach to this issue. They have claimed it is obvious, duh, that maths is true, objective, independent of human minds.

So it is useful to get to the root of the issue.

A second more important reason is that once it is understood that axioms involve choices, then we can consider how those choices are typically made. What is the thought process by which good and useful axioms have been developed? This opens up a discussion of the theory of axiom construction. Whereas if you believe axioms are found rather than developed, then such a conversation, such an epistemological self-examination, seems pointless.

 

[Hurkyl]

But for the moment, the focus has to be on the status of axioms. Can they too really be something we know to be objectively true in "a no choice about it" fashion?

The thing I want to emphasize is that on the mathematical side of things, this doesn't really make sense. Tarski's axioms^* for first-order Euclidean geometry have nothing to do with any notion of truth -- they were chosen because they generate the theory of Euclidean geometry.

The point I had intended in my previous post was that none of your questions relate to the mathematical side of things. IMO, your question is a question of science, not a question of philosophy of mathematics.

Euclidean geometry doesn't care whether points are "zero-D objects" or "pure location" or "limits of a process of dimensional constraint" or even "lines"^**. All of these issues are questions of science -- assertions about how the primitive terms of Euclidean geometry correspond to the real world.

*: Tarski's axioms are a first-order rewrite of Hilbert's axioms, which is a set of axioms that update Euclidean geometry to modern standards of rigor.

**: Observing that you can use "point" to refer to a line and "line" to refer to a point was one of the most important topics in classical geometry -- eventually leading to the discovery of the projective plane.

 

[apeiron]

Euclidean geometry doesn't care whether points are "zero-D objects" or "pure location" or "limits of a process of dimensional constraint" or even "lines"^**. All of these issues are questions of science -- assertions about how the primitive terms of Euclidean geometry correspond to the real world.

Still, there is an essential difference between "geometry now does not care..." and "geometry never did have to care...".

Big claims are made for platonism and objective truth. So it is a critical point if subjectivity was actually always the starting point for whatever followed.

In the earliest stage of intellectual development, science, maths and philosophy are really not different activities. People just looked at the world and tried to make more sense of things by generalising. By realising their initial response was "subjective" and they could do better by stepping back to some more "objective" view.

The sad thing about intellectuals today is how they have gone off in their different directions and each claim they have the one true fundamental way. So mathematicians like to claim the status of dealing in objective truth, scientists claim to have the perfect empirical method, philosophers spin in circles no longer contributing to the meta-view but grateful to have jobs in universities and the rosy glow of past glories.

All three should be aspects of the one coherent knowledge-approaching discipline. Which is why foundational issues like "are axioms subjective" become important. It is precisely because they allow "science" - empiricism, prediction and test - back into the game of axiom development.

 

[Pythagorean]

But the problem was - as supported by the quotes in the OP - that many people have been taken an unreasoned approach to this issue. They have claimed it is obvious, duh, that maths is true, objective, independent of human minds.

I think people tend to confuse emotional with human. Mathematics is obviously dependent on human cognition, but also obviously independent of human emotion. Sometimes people think that if you remove emotion, you remove what makes us human, but I think the way we rationalize and compute is very much part of what makes us human too, and we have separated that process out of our overall brain function and called it mathematics.

But math alone isn't very helpful or useful to us (in terms of prediction) until we introduce quality to it (we define our variables or define what we're quantifying).

 

[apeiron]

Tarski's axioms are a first-order rewrite of Hilbert's axioms, which is a set of axioms that update Euclidean geometry to modern standards of rigor.

Yes, I like Tarski because it is a Peircean, category theory, triadic, systems-style approach. It reduces things even further to points (the maximally constrained) and their generalised (unconstrained) relations. Then there is the thirdness of what relations emerge as self-consistent across such a universe of points. It is exactly my kind of approach to things.

But, it is still possible to imagine other choices here.

For example, the idea of the point is still a spatial one. A more general foundational idea could arguably be spatiotemporal - where what is being maximally located is not just spatial position, but also temporal.

So, in this view, more fundamental than the spatial point is the spatiotemporal event (or instance, or occasion). Now what would geometry look like based on the relations that emerge from a universe of events or occasions? Subjectively, is this an even more fundamental starting point?

Of course, it is hard enough cashing out much of real mathematical value from Tarski's formulation. It is criticised as a more impractical level of modelling, is it not?

Like topology vs geometry, the modelling can become more general and also less useful for the (subjective) purposes of people.

 

[qsa]

Reasonable comments. But the problem was - as supported by the quotes in the OP - that many people have been taken an unreasoned approach to this issue. They have claimed it is obvious, duh, that maths is true, objective, independent of human minds.

So it is useful to get to the root of the issue.

A second more important reason is that once it is understood that axioms involve choices, then we can consider how those choices are typically made. What is the thought process by which good and useful axioms have been developed? This opens up a discussion of the theory of axiom construction. Whereas if you believe axioms are found rather than developed, then such a conversation, such an epistemological self-examination, seems pointless.

Shifting the attention from the objectivity of the contents of axioms to the "choice" of axioms has created some confusion. Or maybe I just misunderstood.


Nevertheless, in fact, the axioms choosen "have no choice" about it, because as we scan the land scape of truths we only pick the most fundamental ,all redundunt facts and facts that do not add anything are neglected. All possible combinations are studied, also new axioms are developed all the time in hope they give us more conclutions and insights. In the end, we are interested in proving higher theorms and make more discovries and find interesting connections. no effort has been spared on finding "good axioms". The subjectivity does not follow.

Trying to come up with best axioms to have all encompassing results is the ultimate in objectivity. Scientist cannot do business without blieving in objectivity(of course with interpretation like positivisim and .. so an), philosophers on the other hand, want to keep things in check, which is fine. If philosophers can come up with better alternatives, I don't think scientists will say no. As a matter of fact a lot of scientists are philosophers themselves and have contributed in their field this way.

 

[apeiron]

I think people tend to confuse emotional with human. Mathematics is obviously dependent on human cognition, but also obviously independent of human emotion. Sometimes people think that if you remove emotion, you remove what makes us human, but I think the way we rationalize and compute is very much part of what makes us human too, and we have separated that process out of our overall brain function and called it mathematics.

But math alone isn't very helpful or useful to us (in terms of prediction) until we introduce quality to it (we define our variables or define what we're quantifying).

I don't think generally that emotion~reason makes a good contrast. Nor intuition~reason. Study the brain and you don't find a serious basis for these divides.

But quantity~quality is a dichotomy that has meaning in this debate. We can see that objective is defined by what can be quantified, subjective remains what is qualitative.

Does this make it easier to see that we start in the qualitative and move out towards the quantitative by a system of measurement? Which in conjunction demands a theory of the metric?

We have to invent - axiomatically - a basis of measurement, then go out and measure the world accordingly. In that fashion we develop from qualitative experience to quantitative, from subjectivity to objectivity.

So we invent the metric or the zero-D point, the rational integer 1. And immediately we put all the measurements that follow on the same quantitative footing.

However - and this is the key point of Rosen's modelling relations epistemology, the best approach I have come across - measurements are the informal part of the process. They are made from the position of subjectivity. As in science, we have to make active choices about when measurements are "good enough". We get into matters like statistical validity.

This is why science is much more aware of its essential subjectivity because it is a factor to be considered in every observation, every experiment, every measurement act.

But the formal modelling part of the story, the theory of the metric, just needs that first subjective choice to get started, then the consequences of "metrification" can be treated as pure quantification activity.

This is why mathematicians, and analytic philosophers, can come to feel that any qualitative aspects of their researches have long been put behind them. Indeed, what they do is actually "objective" and quite a different thing to muddy old science.

Thanks. I think quality~quantity is a very useful distinction in this thread.

 

[Pythagorean]

I don't think generally that emotion~reason makes a good contrast. Nor intuition~reason. Study the brain and you don't find a serious basis for these divides.

I didn't mean to imply a contrast between the two. My point was that people who make the arguments you quoted in the OP may be falsely associating the idea of objectivity with non-emotional... and well, math seems to be non-emotional.

So they think "math is a way to avoid emotional interference! It's objective! It must be independent of human thought!" Which is erroneous thinking, in my opinion.

 

[apeiron]

So they think "math is a way to avoid emotional interference! It's objective! It must be independent of human thought!" Which is erroneous thinking, in my opinion.

Yes. And then they also pull the other stunt that when things are "beautiful" they must be true.

'"Beauty is truth, truth beauty," - that is all/Ye know on earth, and all ye need to know.' said Keats.

And in googling for that quote, I came across this neat little essay-ette on wiki...
http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

Where Mathematics Comes From culminates in a perspective from the cognitive science of mathematics, a case study of Euler's identity as an example - the authors George Lakoff and Rafael E. Núñez argue that this identity reflects a cognitive structure peculiar to humans or to their close relatives, the hominids. Variations of that argument were hinted at by "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" in which Eugene Wigner argued to "abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species." In other words, he suggested that mathematics would not only be perceived differently, it would possibly be quite different in concept for beings of greater or lesser intelligence - more or less what Lakoff and Nunez argued more explicitly forty years later. Rather than refer to beauty, status or absolute truth, Lakoff, Nunez and Wigner all thought advanced mathematics - and thus the identity which is so basic to it - reflected something about specifically human (or primate or mammal or biped) intelligence that might or might not be shared by creatures of very different bodies and minds. Kurt Goedel had even earlier argued that an alien tradition of mathematics might not necessarily rely so much on the consistency or proof concepts used in modern Western mathematics, and that even basic ideas (like the identity) might be different, particularly if there was no foundational reason outside mathematics to believe them. To these thinkers, either the identity shapes the human mind, or the human mind shapes the identity, or they have a symbiotic relationship, one that would explain why beauty or truth are so readily seen in the identity by mathematicians, who are human, and perhaps also why some feel compelled to write bad poetry about it.

 

[apeiron]

There is also this useful criticism of Lakoff/Nunez "maths is subjective" argument...so the counter-argument for why the objects of maths might indeed be objectively real...

http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From

In set theories such as Zermelo-Fraenkel one can indeed have {1,2} = (0,1), as these are two different symbols denoting the same object. The claim that there is an anomaly because these are "fully distinct concepts" is on the one hand not a clear scientific statement, and on the other hand, is on par with such statements as: ""The positive real solution of x2 = 2" and "" cannot be equal because they are fully distinct concepts.".
The apparent anomaly stems from the fact that Lakoff and Núñez identify mathematical objects with their various particular realizations. There are several equivalent definitions of ordered pair, and most mathematicians do not identify the ordered pair with just one of these definitions (since this would be an arbitrary and artificial choice), but view the definitions as equivalent models or realizations of the same underlying object. The existence of several different but equivalent constructions of certain mathematical objects supports the platonistic view that the mathematical objects exist beyond their various linguistical, symbolical, or conceptual representations.
As an example, many mathematicians would favour a definition of ordered pair in terms of category theory where the object in question is defined in terms of a characteristic universal property and then shown to be unique up to unique isomorphism (this was recently mentioned in an article on mathematical platonism by David Mumford).
The above discussion is meant to explain that the most natural and fruitful approach in mathematics is to view a mathematical object as having potentially several different but equivalent realizations. On the other hand, the object is not identified with just one of these realizations. This suggests that the intuitionistic idea that mathematical objects exist only as specific mental constructions, or the idea of Lakoff and Núñez that mathematical objects exist only as particular instances of concepts/metaphors in our embodied brains, is an inadequate philosophical basis to account for the experience and de facto research methods of working mathematicians. Perhaps this is a reason why these ideas have been met with comparatively little interest by the mathematical community.