This is actually similar to a kind of reasoning i have undertaken, but I want to ask you what you make of the fact that such high level abstraction even has any kind of utility at all. Say one day we as all physicalists (assumption on my part) all sit down (in present day mind, but without any knowledge of abstract maths), and we do the same type of physical formal kind of thinking that Newton or Leibniz undertook and developed calculus, which gives us the underpinning of most of modern applied mathematics. We then see how the kind of geometrical reasoning we have employed is able to tap into the ‘world logic’ (for want of a better phrase). If I were to come up to you, describe how we can obtain an enormous amount of world logic insight by some utterly far removed and esoteric notions concerning topological spaces, sobolev spaces and understanding complex abstract interactions in banach spaces (or what have you), why would you, as a physicalist, take that seriously? ie: you can believe there is more work to be done, but that it is very very unlikely to come from understanding concepts that have little or (very often) NOTHING to do with the real world. (indeed this can go quite far, see: https://ncatlab.org/nlab/show/differential+equation)
The core of what I’ve been struggling with is to produce a sufficient response that dissolves that question satisfactorily, and although there must be a connection between mathematics and neural structures (there is a good amount of literature as to this, echoing your own thoughts), I feel like this question is actually quite far removed from that even being the case or not.
With your point b) we can recognise it has order within itself, but then when it comes to that next step, that seeing what structures work on top of it, that process is either (very broadly speaking) an act of creation by the person (leaving out the nature of that act, lets consider it neural) or we simply have insight into another category of the world (platonic), and then we ask the question: ‘If it is neural, why SHOULD this extra addition work? Reality has no means of simply abiding by logical structure or bending itself to fit into abstract reasoning. But now, given that it does work, how are we to interpret the nature of our two choices?’
My point is the process of maths is (to a degree) invented or discovered, and under the invented hypothesis, where one would adhere to strict physicalist-style nominalism, the very act of predicting that the solutions to very real problems are dependent on abstract insight is literally incompatible with that position, to the point where seeing it done, even once, forces you to make some drastic ramifications to your own ontological model of the world.
If I’ve messed anything up, or if you feel I haven’t seen your point, any response would be appreciated.
If I’ve messed anything up, or if you feel I haven’t seen your point, any response would be appreciated.
I think we’re on the same page. Your question is something like ‘why do we live in world that’s more like B than A?’
Story A:
[Physics] is a really cool bunch of fields! It’s done all these really big things!
It’s all because we were able to take all this data, and (eventually) figure out a lot about our universe, at a bunch of different scales. We’re still working on applying some of the information, and we’re working hard to get more data (on really small things, and really big things). We’re also putting a little effort into refining our existing theories more, but not much because we figure future changes in that area will be small, without more data, and the more abstract things are the less they apply to (and are useful in) reality.
Story B:
[Physics] is a really cool bunch of fields! It’s done all these really big things!
You’d think that we’d need lots of data to make all these discoveries, but somehow all the deep truths about the workings of the universe can be deduced if you think about numbers long enough. Weird, huh?
My point is the process of maths is (to a degree) invented or discovered, and under the invented hypothesis, where one would adhere to strict physicalist-style nominalism, the very act of predicting that the solutions to very real problems are dependent on abstract insight is literally incompatible with that position, to the point where seeing it done, even once, forces you to make some drastic ramifications to your own ontological model of the world.
One account is that the particular grouping of features into a definition is “invented”, in the same way that the concept of a “tree” is invented; but there is still a pattern in the world corresponding to tree. But from your original post I think we’re in agreement on this point?
the mathematics says that ‘if we assume such and such holds about, say partial differential equations, then this implies, through a chain of abstract reasoning, we get something like a concrete result within the real world.’ Its essentially this process that disturbs me. This process can fail in the real world, especially when we try to model specific phenomena, but those models have to obey PDE conditions (from our example), and this includes both right and wrong models.
I believe Pattern’s reasoning above could be summed-up by saying that abstraction is a way for us to model the real world, and the process of reasoning abstractly a way for us to run some sort of efficient simulation with our models. (@Pattern Is that a fair one-line summary?)
In which case, my understanding of your original question is one of the two: why is it the case that the world could be *efficiently* simulated? Perhaps your question is even one level deeper: why is it the case the the world could *simulated* at all? After all, it is possible that the only way to predict the outcome of a physical process is to observe the physical process. (Is this a fair summary of what disturbs you about the PDEs example?)
This could be rephrased slightly more concretely as a question about the Church-Turing thesis: how come there is such a thing as a *universal* Turing machine. Made even more concrete, it turns into a deep physics problem: what kind of laws of physics permit the existence of a universal Turing machine. That’s a deep (and technical!) question, which in this particular form was popularized by David Deutsch. This blog post by Michael Nielsen is a good general-audience introduction.
Close enough. (And if your simulation doesn’t match what happened then at least one of these three things is incorrect: your observations, your model, your simulation (you multiplied instead of adding or something).)
In which case, my understanding of your original question is one of the two: why is it the case that the world could be *efficiently* simulated?
It can’t—in its entirety. Abstraction is “the process of removing something”—by removing a lot of things, sometimes what is left can be “efficiently simulated”.
For the sake of discussion I would like to clarify that I regard those ‘structures’ that might be described by different themes of abstract mathematics as the objects that are to be considered either platonic or not. So, platonism in regards to the abstract structures, not necessarily the instantiations of those structures (ie: the example ‘P.D.E structure’ could be represented equally well with either categories or sets as the lifeblood of each respective formulation). So I think I am in agreement as to the grouping of features into ‘invented’ that you detailed earlier, which leaves the pattern still independent.
I think you do actually touch upon a related source of my disturbance (the primary one being, why does the abstract have literally any utility at all?), but perhaps i can re-phrase it as thus: One can engage in quantum mechanics, and it can often be the case that one encounters the only solution to a problem in QM is by using C* algebras or the other abstract analysis ideas. We can talk of using abstraction as a means of modelling the physical universe, but I do not believe that entails that that abstraction in question should even be likely to have utility (indeed, I think the opposite!). Summing it up after following your lead, ‘why is it necessary that to produce a GOOD simulation of the physical universe, you require the existence of non-physical entities like those routinely encountered in mathematics?’
I am not well-versed enough in theoretical computer science to comment upon church-turing machines. I do not mind having a look, but as a default setting I remain doubtful that they will satisfy me, or dissolve my quandary sufficiently, as they can be interpreted via abstract concepts in their formation.
This is actually similar to a kind of reasoning i have undertaken, but I want to ask you what you make of the fact that such high level abstraction even has any kind of utility at all. Say one day we as all physicalists (assumption on my part) all sit down (in present day mind, but without any knowledge of abstract maths), and we do the same type of physical formal kind of thinking that Newton or Leibniz undertook and developed calculus, which gives us the underpinning of most of modern applied mathematics. We then see how the kind of geometrical reasoning we have employed is able to tap into the ‘world logic’ (for want of a better phrase). If I were to come up to you, describe how we can obtain an enormous amount of world logic insight by some utterly far removed and esoteric notions concerning topological spaces, sobolev spaces and understanding complex abstract interactions in banach spaces (or what have you), why would you, as a physicalist, take that seriously? ie: you can believe there is more work to be done, but that it is very very unlikely to come from understanding concepts that have little or (very often) NOTHING to do with the real world. (indeed this can go quite far, see: https://ncatlab.org/nlab/show/differential+equation)
The core of what I’ve been struggling with is to produce a sufficient response that dissolves that question satisfactorily, and although there must be a connection between mathematics and neural structures (there is a good amount of literature as to this, echoing your own thoughts), I feel like this question is actually quite far removed from that even being the case or not.
With your point b) we can recognise it has order within itself, but then when it comes to that next step, that seeing what structures work on top of it, that process is either (very broadly speaking) an act of creation by the person (leaving out the nature of that act, lets consider it neural) or we simply have insight into another category of the world (platonic), and then we ask the question: ‘If it is neural, why SHOULD this extra addition work? Reality has no means of simply abiding by logical structure or bending itself to fit into abstract reasoning. But now, given that it does work, how are we to interpret the nature of our two choices?’
My point is the process of maths is (to a degree) invented or discovered, and under the invented hypothesis, where one would adhere to strict physicalist-style nominalism, the very act of predicting that the solutions to very real problems are dependent on abstract insight is literally incompatible with that position, to the point where seeing it done, even once, forces you to make some drastic ramifications to your own ontological model of the world.
If I’ve messed anything up, or if you feel I haven’t seen your point, any response would be appreciated.
I think we’re on the same page. Your question is something like ‘why do we live in world that’s more like B than A?’
Story A:
[Physics] is a really cool bunch of fields! It’s done all these really big things!
It’s all because we were able to take all this data, and (eventually) figure out a lot about our universe, at a bunch of different scales. We’re still working on applying some of the information, and we’re working hard to get more data (on really small things, and really big things). We’re also putting a little effort into refining our existing theories more, but not much because we figure future changes in that area will be small, without more data, and the more abstract things are the less they apply to (and are useful in) reality.
Story B:
[Physics] is a really cool bunch of fields! It’s done all these really big things!
You’d think that we’d need lots of data to make all these discoveries, but somehow all the deep truths about the workings of the universe can be deduced if you think about numbers long enough. Weird, huh?
One account is that the particular grouping of features into a definition is “invented”, in the same way that the concept of a “tree” is invented; but there is still a pattern in the world corresponding to tree. But from your original post I think we’re in agreement on this point?
I believe Pattern’s reasoning above could be summed-up by saying that abstraction is a way for us to model the real world, and the process of reasoning abstractly a way for us to run some sort of efficient simulation with our models. (@Pattern Is that a fair one-line summary?)
In which case, my understanding of your original question is one of the two: why is it the case that the world could be *efficiently* simulated? Perhaps your question is even one level deeper: why is it the case the the world could *simulated* at all? After all, it is possible that the only way to predict the outcome of a physical process is to observe the physical process. (Is this a fair summary of what disturbs you about the PDEs example?)
This could be rephrased slightly more concretely as a question about the Church-Turing thesis: how come there is such a thing as a *universal* Turing machine. Made even more concrete, it turns into a deep physics problem: what kind of laws of physics permit the existence of a universal Turing machine. That’s a deep (and technical!) question, which in this particular form was popularized by David Deutsch. This blog post by Michael Nielsen is a good general-audience introduction.
Close enough. (And if your simulation doesn’t match what happened then at least one of these three things is incorrect: your observations, your model, your simulation (you multiplied instead of adding or something).)
It can’t—in its entirety. Abstraction is “the process of removing something”—by removing a lot of things, sometimes what is left can be “efficiently simulated”.
For the sake of discussion I would like to clarify that I regard those ‘structures’ that might be described by different themes of abstract mathematics as the objects that are to be considered either platonic or not. So, platonism in regards to the abstract structures, not necessarily the instantiations of those structures (ie: the example ‘P.D.E structure’ could be represented equally well with either categories or sets as the lifeblood of each respective formulation). So I think I am in agreement as to the grouping of features into ‘invented’ that you detailed earlier, which leaves the pattern still independent.
I think you do actually touch upon a related source of my disturbance (the primary one being, why does the abstract have literally any utility at all?), but perhaps i can re-phrase it as thus: One can engage in quantum mechanics, and it can often be the case that one encounters the only solution to a problem in QM is by using C* algebras or the other abstract analysis ideas. We can talk of using abstraction as a means of modelling the physical universe, but I do not believe that entails that that abstraction in question should even be likely to have utility (indeed, I think the opposite!). Summing it up after following your lead, ‘why is it necessary that to produce a GOOD simulation of the physical universe, you require the existence of non-physical entities like those routinely encountered in mathematics?’
I am not well-versed enough in theoretical computer science to comment upon church-turing machines. I do not mind having a look, but as a default setting I remain doubtful that they will satisfy me, or dissolve my quandary sufficiently, as they can be interpreted via abstract concepts in their formation.