Proof In Philosophy
So in tutorials, Jason has stated about a dozen times that you can’t prove anything except in maths, and that in the real world you can only demonstrate/show things to be the case. (Please correct me if that’s wrong, Jason.)
While I agree that you can’t prove many scientific theories, I think there are various statements you can make about the world which can be proven.
An example: the statement “My right hand has a thumb.” (Referring to me.) I can look at my right hand and and yes, it does have a thumb. What’s more I can poke it, feel it, feel other objects with it, hold things with it, bleed from it, trim fingernails from it… etc. As far as I’m concerned, this is absolute proof that my right hand has a thumb.
In fact, I’m just as convinced that my right hand has a thumb as I am that any number of mathematical statements are true. (Ones supported by proofs which I can understand.)
So what’s going on here? Is there really a fundamental distinction between statements of fact about the universe, and statements about maths? And if so, what is it and why? I’d love to hear anyone/everyone’s thoughts on this.
I think you’re right (whoever you are — don’t you want your participation marks for this?!) that the arguments I’ve given in class only apply to scientific theories. Scientific theories have consistently been shown to be wrong throughout history, but that’s not so clearly the case for other theories. OK, good.
There have been lots of other arguments against being certain about anything, but we haven’t talked about them in this course. Anyone who knows some more general philosophy feel like chiming in?
Jason
— Whoops, that was mine. Forgot to write my name at the end.
Angus
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The problem is, that you can’t ever be absolutely certain about any of the evidence of your senses and therefore cannot “prove” anything this way. On the simple end of the scale is the fact that we can make mistakes with sense data - in the case, say, of an optical illusion or even more simply, seeing a stick from a distance and thinking it’s a snake. So on that level we can’t trust this information 100%. Even if we say, under most normal conditions this is unlikely, there are still cases where we may be hallucinating or even dreaming. And finally, even if we want to say that hallucinations and dreams are recognisably different from the experiences of normal waking life (which itself is a difficult claim to make), there’s always the ‘brain in a vat’ scenario. This should be familiar to pretty much everyone through the film The Matrix. Basically the idea is, that since we experience sense-data through the actions of our brains, they could be artificially created through external stimulus, such as if a scientist were keeping our brain in a vat and running a complex ‘simulated world’ program on it. There is no way we could tell the difference between that state of affairs and our reality and therefore we cannot say we are certain of anything, even the existence of our own hands. Of course, this is extreme scepticism and not something most people take on in everyday life, but it does illustrate the difficulty of absolute ‘proof’.
Heather B.
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Ah yes, my worst nightmare is that I am living in my own personal ‘The Truman Show’…
Anyway, I agree with Heather. Because we interpret the world through our senses, we can’t trust this information to be 100% true. I believe this is in part due to biases.
One that comes to mind is attentional bias. This is unknowingly paying more attention to something and paying less attention to others. For instance, some studies have shown that Americans pay less attention to non-verbal cues in communication and more attention to the verbal content of a communication, while it is claimed to be vice versa for Japanese speakers. And there is selective bias, this bias that refers to how people see things as they expect them to. And probably most important for me as a future researcher, there is confirmation bias - only searching for information that confirm my hypothesis. There are heaps more biases out there (I only talked about a few), but nonetheless I don’t trust data interpreted through our senses to be 100%.
Sally
— Yes, those are good points. If I’m dreaming, or I’m a brain in a vat etc I might not have a thumb, it could just be an illusion.
However, if those scenarios are possible, then we also have to consider that our mathematical proofs are also illusions. As an example, I’ve had dreams where I’ve thought a particular proof made complete sense, only to find on waking up that it was total rubbish. So along the same lines, perhaps our confidence in our mathematical proofs is incorrect, and caused by an inability to reason properly in a ‘dream’.
Hence we should be similarly confidence *(edit: confident) about both the existence of physical objects and the correctness of mathematical proofs.
Thoughts?
Angus
Yes! The same points do apply to mathematical proofs. This is a point that even professional philosophers often miss (or maybe disagree with for good reasons, but I can’t think of any such reasons). So they’re not really proofs either, in my opinion … except that they are, because the terminology of maths says they are. There’s nothing else to call them. But philosophically, I think you’re right.
An ex-student from this course did a really good research project on (among other things) the claim that mathematical proofs are always correct. See http://apache.xeny.net/bunny/philosophy-of-maths/Parsonage%20-%20Philosophy%20of%20Mathematics.pdf
Jason
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I’ve wondered about that for mathematical proofs. But what is philosophy’s stand on mathematical axioms?
Sally
Great question!
What most philosophers think about mathematical axioms (I think) is that it’s up to mathematicians to decide what they should be, i.e. it’s not something that philosophers have to worry about. After all, an axiom can’t be either true or false, since it’s just a definition, not a claim. Although sometimes philosophers do worry about the fact that mathematicians are so good at coming up with axioms that turn out to be so useful. What philosophers mostly worry about instead is (a) preaxiomatic ideas in maths, e.g. the nature of numbers, and (b) how mathematicians make inferences, i.e. logic and stuff like that.
Personally, I think philosophers should worry much more about mathematical axioms. If you look at what mathematicians actually do, a LOT of their effort goes into choosing definitions. It might be true that a definition can’t be true or false, but it can be good or bad. I don’t have anything clever to say about it, but I do think we should be studying it.
Jason
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I think the point about mathematical proofs or axioms is that they’re supposed to be considered as necessary truths, and so there is actually no way in which we could be mistaken about them. I don’t know enough complex mathematics to get into anything tricky, but it’s hard to imagine genuinely believing other than 2+2=4 or triangles having three sides. Largely, I guess, because these are truths of definition. What the question then becomes is whether we can ever build anything useful from these without any outside assumptions. Descartes seemed to think we couldn’t and so used the whole thing as evidence for God as provider of knowledge. There are possibly more logical responses.
Heather B.
On whether we can be wrong about mathematical issues, see my ex-student’s project linked to above.
Jason
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I found that a little too mathematical for me to get through easily :/ I guess though, there are two points under discussion here: 1. Whether it is possible for us to be wrong about logical truths 2. Whether mathematical axioms etc. fall into this category anyway. From what I understood of that paper, it seemed to be primarily addressing the second point?
Heather B.
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Yes, good. I agree with you that it’s important to make that distinction (and you already were making that distinction — sorry that I didn’t pick up on it explicitly). And yes, Hugh’s paper is addressing the second point.
One important issue related to the first point is that there are many competing versions of logic, so either (a) one of them is right but we don’t (yet?) know which one, or (b) more or less than one of them is right. No matter which of these options you take, I think you have to conclude that we can be wrong about logical truths, at least at the moment.
The only way out of that would be to say that there is only one right logic, and we already know which one it is. I don’t think that’s even remotely tenable. One reason (not the only one) is that there are paradoxes that nobody knows how to deal with yet. See for example http://xeny.net/PenguinsRuleTheUniverse.
You might like to look at http://consequently.org/papers/pluralism.pdf, or at least the first couple of pages.
Jason
— Doesn’t the fact of a thumb have to match a criteria and therefore there are assumptions made about it?
When thinking about the existence of a thumb as it is explained above it is based on perspective and therefore we have to understand the credibility of the perspective. Although couldn’t we get to a point where we question the existence of even us, despite knowing that we are in existence?
Even though because of our biological constraints of we can rely on our sense for the absolute proof, this can definitely be extend towards the inability to understand the absolute proof in any branch of science.
Although in saying this, i think there is much more emphasis on the actual explanation of proving it then the the visibility proof.
Bernadette
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We likely have to consider Kant’s distinction between phenomena (things as they appear to the observer), and noumena (things in themselves, which essentially constitute reality). We have no empirical access to the “real” world, if you will, as these two things are different, and thus constitute separate worlds. We can only perceive through our senses, which are fallible, so we have no knowledge of how things really are beyond our impressions of them. Following this, visible proof doesn’t exist and we can never have true knowledge.
Emily Haag
Emily, yes, excellent point. It’s not the only reason to be dubious about proof, but it is an important one. See also Hypotheses - Instrumentalistic Theories Accepted By The Church Doctrine, and also I’m going to talk about Kant very briefly in a future lecture.
