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Proofs, Implications, and Models
Followup to: Causal Reference
From a math professor's blog:
One thing I discussed with my students here at HCSSiM yesterday is the question of what is a proof.
They’re smart kids, but completely new to proofs, and they often have questions about whether what they’ve written down constitutes a proof. Here’s what I said to them.
A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.
This is not quite the definition I would give of what constitutes "proof" in mathematics - perhaps because I am so used to isolating arguments that are convincing, but ought not to be.
Or here again, from "An Introduction to Proof Theory" by Samuel R. Buss:
There are two distinct viewpoints of what a mathematical proof is. The first view is that proofs are social conventions by which mathematicians convince one another of the truth of theorems. That is to say, a proof is expressed in natural language plus possibly symbols and figures, and is sufficient to convince an expert of the correctness of a theorem. Examples of social proofs include the kinds of proofs that are presented in conversations or published in articles. Of course, it is impossible to precisely define what constitutes a valid proof in this social sense; and, the standards for valid proofs may vary with the audience and over time. The second view of proofs is more narrow in scope: in this view, a proof consists of a string of symbols which satisfy some precisely stated set of rules and which prove a theorem, which itself must also be expressed as a string of symbols. According to this view, mathematics can be regarded as a 'game' played with strings of symbols according to some precisely defined rules. Proofs of the latter kind are called "formal" proofs to distinguish them from "social" proofs.
In modern mathematics there is a much better answer that could be given to a student who asks, "What exactly is a proof?", which does not match either of the above ideas. So:
Meditation: What distinguishes a correct mathematical proof from an incorrect mathematical proof - what does it mean for a mathematical proof to be good? And why, in the real world, would anyone ever be interested in a mathematical proof of this type, or obeying whatever goodness-rule you just set down? How could you use your notion of 'proof' to improve the real-world efficacy of an Artificial Intelligence?
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Counterfactual resiliency test for non-causal models
Non-causal models
Non-causal models are quite common in many fields, and can be quite accurate. Here predictions are made, based on (a particular selection of) past trends, and it is assumed that these trends will continue in future. There is no causal explanation offered for the trends under consideration: it's just assumed they will go on as before. Non-causal models are thus particularly useful when the underlying causality is uncertain or contentious. To illustrate the idea, here are three non-causal models in computer development:
- Moore's laws about the regular doubling of processing speed/hard disk size/other computer related parameter.
- Robin Hanson's model where the development of human brains, hunting, agriculture and the industrial revolution are seen as related stages of accelerations of the underlying economic rate of growth, leading to the conclusion that there will be another surge during the next century (likely caused by whole brain emulations or AI).
- Ray Kurzweil's law of time and chaos, leading to his law of accelerating returns. Here the inputs are the accelerating evolution of life on earth, the accelerating 'evolution' of technology, followed by the accelerating growth in the power of computing across many different substrates. This leads to a consequent 'singularity', an explosion of growth, at some point over the coming century.
Before anything else, I should thank Moore, Hanson and Kurzweil for having the courage to publish their models and put them out there where they can be critiqued, mocked or praised. This is a brave step, and puts them a cut above most of us.
That said, though I find the first argument quite convincing, I find have to say I find the other two dubious. Now, I'm not going to claim they're misusing the outside view: if you accuse them of shoving together unrelated processes into a single model, they can equally well accuse you of ignoring the commonalities they have highlighted between these processes. Can we do better than that? There has to be a better guide to the truth that just our own private impressions.
The bias shield
A friend asked me to get her Bill O'Reilly's new book Killing Lincoln for Christmas. I read its reviews on Amazon, and found several that said it wasn't as good as another book about the assassination, Blood on the Moon. This seemed like a believable conclusion to me. Killing Lincoln has no footnotes to document any of its claims, and is not in the Ford's Theatre national park service bookstore because the NPS decided it was too historically inaccurate to sell. Nearly 200 books have been written about the Lincoln assassination, including some by professional Lincoln scholars. So the odds seemed good that at least one of these was better than a book written by a TV talk show host.
But I was wrong. To many people, this was not a believable conclusion.
(This is not about the irrationality of Fox network fans. They are just a useful case study.)
Paper: Testing ecological models
You may be interested in a paper of medium age I just read. Testing ecological models: the meaning of validation (PDF) tackles a problem many of you are familiar with in a slightly different context.
To entice you to read it, here are some quotes from its descriptions of other papers:
Holling (1978) pronounced it a fable that the purpose of validation is to establish the truth of the model…
Overton (1977) viewed validation as an integral part of the modelling process…
Botkin (1993) expressed concern that the usage of the terms verification and validation was not consistent with their logical meanings…
Mankin et al. (1977) suggested that the objectives of model-building may be achieved without validating the model…
I have another reason for posting this; I’m looking for more papers on model validation, especially how-to papers. Which ones do you consider most helpful?
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