Let x be arbitrary, and let ε > 0. Then we pick δ = ε/2. If |y-x| < δ, then |f(y)-f(x)| = |2y-2x| = 2|y-x| < 2δ = ε. Therefore f is continuous at x, and consequently at every point since x was arbitrary.
It seems to me that a disanalogy between the mathematical analysis of continuity and this philosophical analysis of agency is that now mathematicians no longer care about common usage of the word “continuous”. The epsilon-delta definition *is* continuity now. Is this also the case for philosophical analyses? If they were shown to have clear divergences from common usage of the concepts being analyzed, would the analyses be discarded?
As a professional mathematician (grad student) i would say that proving what follows from axioms and definitions is at least a lot if not all of math. Choice of Axioms (hey, that’s my substack) is not something we explicitly concern ourselves with, but i would add that the primary focus of the job is *concept construction* and not axiomatics.
I don’t know of many mathematicians who could confidently tell you all the Zermelo-Fraenkel axioms. Mostly we work with constructed concepts (real numbers, functions, operators, etc.) without bothering with the underlying axioms.
I don’t know if what I’m saying is clear and i apologise; I’m still early in my career and still trying to make sense of it all!
I conjecture you might be a fan of Will Ratoff's work. He's been arguing that Fristonian ideas about predictive processing in neuroscience vindicate a picture of practical reasoning very much like the one you're describing here (he also draws the link to Velleman):
Do you think this view has ramifications for what an account of belief has to look like? At least on the surface, it looks like e.g. a dispositionalist account of belief could be circular (action is analysed in terms of belief, while belief is analysed in terms of action).
I kinda touch on a concern like this in footnote 5.
1. I believe but am not super duper confident that the conditions can give an analysis of an action, mainly because of the concern that some terms therein might be defined in terms of action. (I am actually attracted to a non-dispositionalist view that defines belief in terms of action, but my thoughts are not sorted out). Nevertheless, a claim of the form “an action is a belief satisfying such-and-such conditions” would still be useful if it does not count as a definition.
2. I don’t really think beliefs etc. are reducible to physical stuff. Not in a spooky way—I can grant that all there is is physical stuff, but we might have *concepts* that are not reducible to physical stuff. Instead, in the case of beliefs, I think we get our concept of a belief from our self-consciousness of our own activity as believing agents, not from observing the empirical conditions under which agents have beliefs.
The British writer cp snow would love this piece! You are valuing both mathematical and literary sophistication. Both will expand your mind but not everyone is going to be equally skilled at both.
By the pencil definition of continuity curves with vertical lines in them (as in say an function which includes a Heaviside step function say) could be continuous which would be wrong (I think I have a BS in physics so clearly I can't be trusted about math). Also this suggests to me any formula with multiple solutions (so for a given x more than 1 y value) will fail to be continuous on this definition, although you could break a two value formula into two separate continuous functions.
There is the dispositional account of belief where the content of a belief just is a disposition to act. The thing linking my willingness to respond in English that "Paris is the capital of France" and au Francais que "Paris c'est la capitale de France" is my belief that Paris is the capital of France, suggesting that belief is not a linguistic proposition but something else, why not a disposition to act that makes one appropriately disposed to act in different circumstances (my vague sense would be this move has several problems, including not using some circular construction like belief in X is the disposition to act as if X is true, but it still has its defenders). If someone makes a version of that that really works I feel like its going to be more natural that in order to act in a certain way you must form a disposition to act that includes that particular act under the umbrella, rather than acting a certain way causes you to gain that disposition, though there might be plausible examples of that.
Also I would have thought only justified beliefs required evidence, I thought beliefs in general could just pop into your head via any sort of mechanism such as moved by the animal spirits, due to a blow to the head etc.
In order to be a continuous function, something must be a function first. At any rate if you are dead set on making the pencil definition work for these cases you can consider a curve, that is a function f: [0, 1] -> R^2, which you can imagine as a trajectory parametrized by time.
The step function and its cousin the impulse function are both great examples of discontinuous functions that are tremendously useful in real life. Try to model an electrical circuit or understand DSP without them!
I am completely unfamiliar with philosophy of action (sorry) but I have some questions about your view. My scenario would be something like the following (which I've realised looks a bit like a Frankfurt case):
You have been wired up to some machine in such a way that at a certain point if you try to lift your arm nothing will happen, but if you don't try to, then the machine will cause your arm to move involuntarily. Say also that you are aware of this state of affairs. You know that your arm will raise either way.
You have the belief that your arm will raise at a certain time but nevertheless it seems to me that you can still choose to try to raise your hand or not. I can imagine on the one hand having the ordinary experience of desiring that my hand raises, and having it occur by ordinary means without machine intervention. I can also imagine desiring that it doesn't but knowing that it will either way, and having the machine make it happen.
Is your view committed to saying that in this scenario I actually can't act? If so what would distinguish the two cases, if anything? Is my framing of the scenario even coherent under your view?
Great first half, more math in philosophy would be nice. Especially when people apply concepts like complexity or probability.
I'm not convinced by the second half, though. I don't see why it's"(i) that you never have a belief that is not supported by sufficient evidence" and not "(iii) that you never form a belief that is not supported by sufficient evidence" - having time variable doesn't seem unreasonable.
Third condition also seems a bit like an easy way to get out of any situation this would potentially fail, because of how vague "psychologically constituted" is.
In case any of you want to check your work:
Let x be arbitrary, and let ε > 0. Then we pick δ = ε/2. If |y-x| < δ, then |f(y)-f(x)| = |2y-2x| = 2|y-x| < 2δ = ε. Therefore f is continuous at x, and consequently at every point since x was arbitrary.
It seems to me that a disanalogy between the mathematical analysis of continuity and this philosophical analysis of agency is that now mathematicians no longer care about common usage of the word “continuous”. The epsilon-delta definition *is* continuity now. Is this also the case for philosophical analyses? If they were shown to have clear divergences from common usage of the concepts being analyzed, would the analyses be discarded?
If philosophers had a better understanding of what they were supposed to be doing, the reactions would be analogous
As a professional mathematician (grad student) i would say that proving what follows from axioms and definitions is at least a lot if not all of math. Choice of Axioms (hey, that’s my substack) is not something we explicitly concern ourselves with, but i would add that the primary focus of the job is *concept construction* and not axiomatics.
I don’t know of many mathematicians who could confidently tell you all the Zermelo-Fraenkel axioms. Mostly we work with constructed concepts (real numbers, functions, operators, etc.) without bothering with the underlying axioms.
I don’t know if what I’m saying is clear and i apologise; I’m still early in my career and still trying to make sense of it all!
I conjecture you might be a fan of Will Ratoff's work. He's been arguing that Fristonian ideas about predictive processing in neuroscience vindicate a picture of practical reasoning very much like the one you're describing here (he also draws the link to Velleman):
https://williamjeratoff.weebly.com/my-work.html
Do you think this view has ramifications for what an account of belief has to look like? At least on the surface, it looks like e.g. a dispositionalist account of belief could be circular (action is analysed in terms of belief, while belief is analysed in terms of action).
I kinda touch on a concern like this in footnote 5.
1. I believe but am not super duper confident that the conditions can give an analysis of an action, mainly because of the concern that some terms therein might be defined in terms of action. (I am actually attracted to a non-dispositionalist view that defines belief in terms of action, but my thoughts are not sorted out). Nevertheless, a claim of the form “an action is a belief satisfying such-and-such conditions” would still be useful if it does not count as a definition.
2. I don’t really think beliefs etc. are reducible to physical stuff. Not in a spooky way—I can grant that all there is is physical stuff, but we might have *concepts* that are not reducible to physical stuff. Instead, in the case of beliefs, I think we get our concept of a belief from our self-consciousness of our own activity as believing agents, not from observing the empirical conditions under which agents have beliefs.
The British writer cp snow would love this piece! You are valuing both mathematical and literary sophistication. Both will expand your mind but not everyone is going to be equally skilled at both.
By the pencil definition of continuity curves with vertical lines in them (as in say an function which includes a Heaviside step function say) could be continuous which would be wrong (I think I have a BS in physics so clearly I can't be trusted about math). Also this suggests to me any formula with multiple solutions (so for a given x more than 1 y value) will fail to be continuous on this definition, although you could break a two value formula into two separate continuous functions.
There is the dispositional account of belief where the content of a belief just is a disposition to act. The thing linking my willingness to respond in English that "Paris is the capital of France" and au Francais que "Paris c'est la capitale de France" is my belief that Paris is the capital of France, suggesting that belief is not a linguistic proposition but something else, why not a disposition to act that makes one appropriately disposed to act in different circumstances (my vague sense would be this move has several problems, including not using some circular construction like belief in X is the disposition to act as if X is true, but it still has its defenders). If someone makes a version of that that really works I feel like its going to be more natural that in order to act in a certain way you must form a disposition to act that includes that particular act under the umbrella, rather than acting a certain way causes you to gain that disposition, though there might be plausible examples of that.
Also I would have thought only justified beliefs required evidence, I thought beliefs in general could just pop into your head via any sort of mechanism such as moved by the animal spirits, due to a blow to the head etc.
In order to be a continuous function, something must be a function first. At any rate if you are dead set on making the pencil definition work for these cases you can consider a curve, that is a function f: [0, 1] -> R^2, which you can imagine as a trajectory parametrized by time.
The step function and its cousin the impulse function are both great examples of discontinuous functions that are tremendously useful in real life. Try to model an electrical circuit or understand DSP without them!
Thanks, that was an interesting read.
I am completely unfamiliar with philosophy of action (sorry) but I have some questions about your view. My scenario would be something like the following (which I've realised looks a bit like a Frankfurt case):
You have been wired up to some machine in such a way that at a certain point if you try to lift your arm nothing will happen, but if you don't try to, then the machine will cause your arm to move involuntarily. Say also that you are aware of this state of affairs. You know that your arm will raise either way.
You have the belief that your arm will raise at a certain time but nevertheless it seems to me that you can still choose to try to raise your hand or not. I can imagine on the one hand having the ordinary experience of desiring that my hand raises, and having it occur by ordinary means without machine intervention. I can also imagine desiring that it doesn't but knowing that it will either way, and having the machine make it happen.
Is your view committed to saying that in this scenario I actually can't act? If so what would distinguish the two cases, if anything? Is my framing of the scenario even coherent under your view?
Sorry if I've missed anything obvious.
Great first half, more math in philosophy would be nice. Especially when people apply concepts like complexity or probability.
I'm not convinced by the second half, though. I don't see why it's"(i) that you never have a belief that is not supported by sufficient evidence" and not "(iii) that you never form a belief that is not supported by sufficient evidence" - having time variable doesn't seem unreasonable.
Third condition also seems a bit like an easy way to get out of any situation this would potentially fail, because of how vague "psychologically constituted" is.
I’m on social security. No way I’m gonna get any amount of money in 10 minutes dude 😆