This is something I been thinking about for a while and I would appreciate the insights of anyone with experience working with inutition.
Especially with the rise of the X-Phi movement there has been an increased interest in philosophical and psychological intuition, particularly with regards to whether or how arguments from intuition should be used as evidence in philosophical discourse. There has been much less attention paid in recent years to the reliability/usefulness of mathematical intuition, though it’s use is similarly widespread. The question then, is whether philosophical and mathematical intuitions have similar epistemic import, and if so, how much?
The most obvious case for the use of intuition in both philosophical and mathematical arguments is that it has been relied on, in both fields, for millennia — I would like to meet the philosopher or mathematician who would deny that progress has been made in either field, or at even that intuition has never played a role. Indeed, mathematicians and philosophers of mathematics often appeal to the intuitive plausibility of the axioms of ZFC; and it would be difficult to deny that our intuitions about barn facade land and stopped clocks have not played a significant role in the shaping of contemporary epistemology.
On the other hand, there are numerous empirical and mathematical results have been taken to show that intuition is inherently unreliable. You would be hard pressed to find an empirical study where 100% of the participants have identical intuitions. Likewise, one can list counterintuitive mathematical results seemingly ad nauseum — plane filling curves, nowhere differentiable functions, three `countries’ that share all of their borders…
So far so similar.
Another point in favour of treating philosophical and mathematical intuitions similarly is in their positive uses: in both disciplines intuitions are used to support the premises (axioms) of arguments (proofs), or as a starting point for research.
The biggest difference, I submit, is in in the negative use of intuition. If a philosophical theory has unintuitive results, the proponents of said theory will generally be called upon to provide a compelling argument as to why such consequences should be accepted. In mathematics however, unintuitive results are generally taken as evidence that our intuitions were incorrect. This seems to be more a difference in methodology than in the nature of the intuitions, and even this difference breaks down if the axioms used by the mathematician are not generally accepted, such as in the case of non-standard set-theories.
So it seems that philosophical and mathematical intuitions are not so different, and thus that evidence for or against the reliability of one may be equally applicable to the other. One must take into account the methodological differences when evaluating intuitions, but in any case this must be done explicitly and thoroughly if arguments from intuition are to be used in any argument.
I will finish with the caveat that all of this depends on exactly how `intuition’ is defined — a difficult and contentious project for sure, but one that would give a much more precise nature to any discussion such as this.
If anyone is interested in reading a longer paper on this topic (I presented a paper on this topic last year at a conference in New England) please feel free to email me aphilosopherstake (at) gmail.com. Thanks in advance for your feedback.
-Aaron
A Philosopher's Take 
brynrwilliams
September 3, 2013
Hi Aaron, just a quick thought. Perfectly appropriate in Philosophy to reply “…all the worse for our intuitions.” to an argument from counterintuitive conclusions. I’m thinking mostly here of Davidson and his response to Swampman. Ultimately, in both Math and Phil, intuitions form, at best, a persuasion to accept a conclusion. That doesn’t make them unimportant, but they will (hostage to misfortune) NEVER trump an empirical or logical proof. They may well give rise to the motivation to test asserted proofs, but they are not (again, I’m very much open to correction) arguments. They do turn out to be very valuable in rhetoric though.
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aaronrtb
September 6, 2013
Although that kind of hand-wavy response is common, I don’t think many philosophers find it satifactory, absent some more robust justification.
I agree that empirical evidence or logical proof should never be hostage to intuition, but in philosophy we often don’t have the option of anything so clear cut. Furthermore, although a mathematical or logical proof should never involve intution, accept as a starting point, axioms and rules of inference are of justified in part by appealing to their intuitive appeal.
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LJ Elkin
September 3, 2013
“The biggest difference, I submit, is in in the negative use of intuition. If a philosophical theory has unintuitive results, the proponents of said theory will generally be called upon to provide a compelling argument as to why such consequences should be accepted. In mathematics however, unintuitive results are generally taken as evidence that our intuitions were incorrect.”
The difference may be due to the higher level of abstraction in mathematics. I find that the role of intuition in epistemology, mind, and ethics is unclear. However, from observations of the discipline, it seems that at times the role of intuition in the latter areas of study is to satisfy psychological preferences. Many don’t want eliminativism or cultural relativism or determinism to be true (I’m not sure what to say about epistemology) because if any one of them turns out to be true, it would be detrimental to ordinary practices, which the thought creates an uneasy feeling. So appealing to “commonly assumed intuitions” (though X-PHI-ers have shown that some of these intuitions are not so common), one can make a case against the non-preferable theses.
In mathematics, however, many don’t have much at stake, prima facie, in the outcome of mathematical research programs. Godel’s incompleteness theorems don’t have a direct impact on our practical lives. So the derivation of theorems leading to the end of certain research programs hadn’t affected the way people practically moved on with their lives. Pure mathematics is of great importance and is foundational to many mathematically driven disciplines, but its abstractness eliminates any bias toward accepting unintuitive results because of the practical disconnect with ordinary life.
As one moves toward the more concrete, the bias begins to emerge. We might think of physics as an in-between area. Physics is highly mathematical, especially quantum physics, but it also has practical application. In quantum mechanics, many unintuitive results have been derived throughout the past century. They are generally accepted now within the scope of the discipline. But when applied to everyday life, there is resistance against accepting the unintuitive results. For example, the Everettian Many-Worlds-Theory is becoming widely accepted by prominent theoretical physicists. The results of the theory are highly unintuitive and if true the theory would significantly impact how we think about the world. Instead of contemplating accepting the unintuitive results, common folk describe it as science fiction and some philosophers employ common sensical examples in order to undermine the results. For a lot of folk, belief in the Many-Worlds-Theory causes disarray in their overall belief systems. Instead of cleaning house in the system of belief, common folk and some philosophers choose to remain steadfast and throw out the newly acquired conflicting evidence.
As Quine had suggested, revising one’s beliefs is more of choice. Ultimately, then, partiality and bias is going to affect one’s choice. In most cases, I think that the decision will be to side with whatever accords with intuition if it so happens that there is something practical at stake. Otherwise, accepting unintuitive results that don’t have a practical impact, like in mathematics, is a less difficult decision.
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Sreejith
September 5, 2013
Hi LJ Elkin,
You said: “I find that the role of intuition in epistemology, mind, and ethics is unclear”. Can you please substatntiate this poin. It will be great if you can tell me in what ways you think the role of intution is not clear in epistemology. (I am aware that there are empirical studies on the Gettier cases to see whether people from different cultural back ground has differnet intuitions. You may have some larger point I presume).
Regards,
Sreejith
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aaronrtb
September 6, 2013
You make a good point about levels of abstraction, I had not looked at it that way, but there is definately something to be said there. As far as Everettianism, it may be unintuitive, but I think for many people it has the appeal of being elegant and explaning the phenoma more satisfactorily (many would say) than Copenhagen. On the other hand it doesn’t seem to be empirically testable, which coupled with intution, is enough for a least skepticism, if not rejection.
It would be interesting to investigate whether scientific intuitions are a mid-point between philosophical and mathematical intuitions, particularly since the methods for testing those intutions will be very different in all three cases.
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LJ Elkin
September 6, 2013
@Sreejith: I hadn’t thought about the role of intuition in philosophy until recently. In many papers that I have written, I have provided arguments from intuition via cases and thought experiments. But now that the role of intuition has become a hot topic of interest in philosophical methodology and metaphilosophy, I tend to closely observe how intuitions are drawn out via cases and thought experiments in philosophical papers. I have started to wonder whether intuitions serve as evidence for or against a claim/theory or if intuitions are meant to preserve commonsense and/or satisfy practical interests. I take the first possibility as truth-seeking. If intuitions count as evidence, it seems that they would raise the probability of a claim being true or false. And the goal would be to identify the truth. I take the second possibility as pragmatic rather than truth-seeking. If intuitions are useful in preserving commonsense and/or satisfying practical interests, then practicality takes priority over Truth.
From observations, I’ve seen both roles employed. Intuitions as evidence seems fairly common and useful in exploring the possibilities extended across conceptual space. However, there are some philosophers (similar to some scientists) that interpret arguments from intuition too strongly. They take a single argument/case/thought experiment as strong evidence that fully confirms a philosophical position while ignoring empirical evidence and/or philosophical counterexamples. This is confirmation bias within philosophy. I am guilty of it as well. I have taken the to time to construct well-crafted cases in epistemology to support a thesis. Many of us have been there. But reflecting on it, I don’t think my motivation was to discover a problem in conceptual space but rather to ultimately support my position. I think that this is true of many philosophers given the detail and carefully worded examples constructed.
The pragmatic role is employed often as well and may even stem from the mentioned confirmation bias. We all hold beliefs about the world and take certain things to be true. Some of those beliefs have practical importance to us. For example, suppose that x believes that moral realism is true. It matters to x that moral realism is true for the reason that she values certain types of actions. The moral realist x finds cultural relativism to be psychologically unsettling in that it does not necessarily accord with her values and preferences. So x rejects the cultural relativist position by using arguments from intuition (e.g. killing is wrong). It seems that some philosophers use intuitions to support their most entrenched beliefs–the beliefs that hold much practical value. And they do so without the aim of truth in mind. But I think that this is explained by the Quinean picture of the entrenched beliefs in the Web of Belief being central and therefore more difficult to abandoned. The use of intuitions, then, may just be an attempt to justify what the valuable central beliefs.
In addressing your general question, I am unclear on whether the evidential role or pragmatic role is the proper role of intuition in philosophical methodology. But maybe both ought to be abandoned in favor of something else. I don’t know and that is where the lack of clarity lies for me with regard to the role of intuition.
In response to your particular question about epistemology, I think that my uncertainty toward the role of intuition, generally speaking, extends to epistemology. Do Gettier cases, knowledge norm examples, skeptical scenarios, etc. provide evidence for/against claims or are they used in favor of supporting a valued epistemological view or do the cases and examples fail to provide much philosophical insight in analyzing knowledge, belief, and justification?
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Rayanna
December 7, 2013
Hi Andrew,I think that cases, which are in themselves inetresting, are weakened by the intuitions which introduce the cases. Intuitions underpinning contents may be thought of as providing cross-modal content, that’s some kind of function they have in supporting thought experiments'(though, of course, just as likely, one’s intuitions are relatively constrained as are one’s normal judgments, as one thinks of them). However one is to have (imagine having) intuitions, are these intuitions subject to the sorts of content/context constraints that would apply in case they were stated? A function of thought experiments may be to indicate the distinction.I’m hunting for the Predelli paper. In it I recall was a discussion of indexicality focused on the distinction between type theorists (Kaplan) and token theorists (Predelli, Reichenbach) as I can’t find the paper the point is kind of eluding me, but it was something to do with wanting to preserve a roughly Russellian or direct level of reference in contexts sufficient to derive comparisons at the level of content.
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LJ Elkin
September 6, 2013
@aaronrtb: Yes, I think that philosophers (and many physicists) find the many-worlds theory more satisfying in its explanatory power than the Copenhagen interpretation. The latter leaves open too many questions regarding uncertainty and indeterminism whereas one could theoretically construct a deterministic framework for the former.
I am not a physicist or mathematician. I can only go by what experts in the field say about the various interpretations of quantum mechanics. But there does seem to be strong consensus for the math working out in many-worlds theory and Copenhagen interpretations (as well as the Bohmian interpretation, for that matter). However, there seems to be resistance from all sides on the implications of the various theories. Thus, there is a gap between empirical content in thought about the structure of the world and mathematical proof.
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Justin Caouette
September 7, 2013
Excellent post and very nice thread. Below is a link to an excellent article comparing mathematical knowledge to moral knowledge. There are lots of intuitions being thrown around in the piece and it seems quite relevant to the discussion.
http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0068.2012.00875.x/pdf
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