Timeline for Defining the standard model of PA so that a space alien could understand
Current License: CC BY-SA 4.0
8 events
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| Sep 20, 2019 at 1:38 | comment | added | Timothy Chow | @TommyR.Jensen : Current deep learning neural nets are astonishingly good at developing the ability to recognize (or approximately recognize) patterns that have not been hard-coded into them. The aliens might not be able to detect "3" with 100% accuracy but they might achieve 99% accuracy. As for whether neural nets have "mental capacity"---that may be a semantic debate about what the word "mental" means. One can imagine that a sufficiently large swarm of ant-like aliens might have enough "mental capacity" with each ant playing the role of (something like) a neuron. | |
| Sep 19, 2019 at 17:47 | comment | added | Tommy R. Jensen | Also, the question itself can only apply to a type of alien which would have the mental capacity required. A reasonable assumption would be "to have the ability to count". Without that requisite, the question becomes moot. Cf the aliens in movies like "Alien" or "Aliens". | |
| Sep 19, 2019 at 17:39 | comment | added | Tommy R. Jensen | I suspect that aliens if they have no prior notion of the number 3, there would be no reason for them to notice whether there are exactly three symbols or not to mark a box with a desirable content. Especially if they automatically perceive a combination of three different symbols as an entirety, or, conversely, if they perceive each symbol to be composed of a variety of subelements. | |
| May 27, 2019 at 17:53 | comment | added | Andreas Blass | @PaceNielsen Before asserting or denying that my mental picture of $\mathbb N$ is the same as yours, I'd want a clear notion of what it means for pictures in two minds (or brains) to be the same. That notion might have something to do with homologous neurons firing in similar circumstances, but I think neuroscience today (and not merely my knowledge of it) is far from making such a notion precise. In other words, I don't even think it makes sense to ask whether two people have the same picture of $\mathbb N$ (or of anything else). | |
| May 23, 2019 at 22:16 | comment | added | Timothy Chow | @PaceNielsen (cont'd): What we can say is that mathematicians who have been suitably trained in the linguistic usage of the term "standard model of the natural numbers" are able to reach agreement about which mathematical inferences employing that phrase are valid/invalid, in exactly the same way that they are able to reach agreement about inferences in any other area of mathematics. The empirical fact that the mathematical community is able to reach a high level of agreement about which inferences are valid is often taken to be a justification of the meaningfulness of mathematical activity. | |
| May 23, 2019 at 17:51 | comment | added | Timothy Chow | @PaceNielsen : Yes, that is a better way of phrasing the question. I would say that whatever it is that we appeal to for justification, it cannot be mathematical proof in the usual sense, because the claim that your internal mental picture matches my internal picture (in some sense at least) is not a mathematical statement. This observation may make some uncomfortable, because mathematicians like to think of themselves as relying solely on "proof" and not on any murky philosophical assumptions. But we are deluding ourselves if we think that mathematics is free of philosophical assumptions. | |
| May 23, 2019 at 16:56 | comment | added | Pace Nielsen | Good points. This reminds me of that old Chinese room problem. Anyway, if it helps get rid of distractions, here is another formulation: You have (presumably) a picture in your head of the natural numbers, as do I. We try to express these pictures using finite symbols, sounds, words, etc... to each other. We seem to agree that we have the same picture. Are we justified in claiming that we in fact do have the same picture? If not, is anyone really justified in using the phrase "standard model of the natural numbers"? If so, what justifies it? | |
| May 23, 2019 at 15:17 | history | answered | Timothy Chow | CC BY-SA 4.0 |