Are axioms subjective?

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  • #26
apeiron
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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.
 
  • #27
Hurkyl
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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.
 
  • #28
apeiron
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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.
 
  • #29
Pythagorean
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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).
 
  • #30
apeiron
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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.
 
  • #31
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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 devoloped 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 alot of scientists are philosophers themselves and have contributed in their field this way.
 
  • #32
apeiron
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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.
 
  • #33
Pythagorean
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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.
 
  • #34
apeiron
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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.
 
  • #35
apeiron
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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.
 
  • #36
apeiron
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  • #38
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they were chosen because they generate the theory of Euclidean geometry.
That doesn't really respond to the problem.
Why is Euclidean geometry important?
Its important to us, no doubt, but why.

All our reasons for creating mathematics the way we have, are based on what mathematicians have wanted to do with math. Its true, once you have your axioms, you can start refining your method.... adding new axioms and delving into the implications of how your axioms interact, but invariably math has to have a purpose before you can start defining axioms.

The purpose is us living our lives within the constraints of our reality. But essentially that is always going to be subjective, based on our personal and collective perspective on things.
 
  • #39
apeiron
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How would you sum up what Putnam is trying to say?

I found that chapter very woolly and rambly - not helped by so many pages being missing of course.

It did start out with the argument that if there are many equivalent descriptions in math, then this should bolster believe that there is something deeper and solid that maths has been trying to describe - a bit like the feeling different parts of an elephant example.

But actually, when you look at all the examples Putnam offers, you find they are dualities and antimonies. Or what I call dichotomies or asymmetries.

And dichotomies are about immanence rather than transcendence. It is the division of the one in two directions, towards complementary limits. So again we would be talking about processes of development rather than "objective" existence.
 

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