Great article! You nailed the two major areas where a purely materialist view breaks down: math and consciousness.
Penrose has a really sophisticated understanding of how these things interweave. I highly recommend the first few chapters of Road to Reality (but the book as a whole is really dense, I had to fight my way through it). Here’s a good summary: https://www.futilitycloset.com/2019/11/12/an-eternal-triangle/
Excellent post! I broadly agree, but have a suspicion that mathematical ‘truths’ are really just tautologies that are true by definition, they just seem profound because our minds are so limited.
I also think materialism is just a shared conscious inference — I think you made this point really well.
In the end, the only thing that we’re really sure of is that consciousness exists, though. This makes me lean towards idealism
Totally agree that mathematical truths are tautologies. But that, in and of itself, is very interesting to me!
A "mathematical truth" isn't true in absentia of axioms and production rules. Once you give the axioms and reduction rules, what you have when considering a "mathematical truth" is really an asymmetric tautology, where one side is really simple (just axioms and production rules) and the other is some theorem that can be derived by operating on the axioms with the production rules. It would be wildly, wildly complex - and yet in the end, it's still 'just' a tautology.
This is what i find to be so cool about mathematical truth; you can start with nothing, pull some rules out of thin air, and then say "X,Y,Z rules lead to theorem P", and _THAT_ is now suddenly real, true, universal, regardless of what the rules of physics are wherever you happen to be.
> In the end, the only thing that we’re really sure of is that consciousness exists, though.
I think i understand this perspective, but i don't share it. I feel sure of other things, but then this makes me realize that a lot of people seem to think 'one ought not feel certain about things unless criterion x is met'. And it seems like a lot of arguments or disagreements ultimately come down to different norms about when it is, or is not, appropriate to feel certain.
This post made my day, Mark! I have suggested it as reading for my local ACX discussion group. Will have to reread and follow all the links, but so far it warms the heart of this middle-aged Catholic convert with a BS in physics at a dark time of year.
Sorry, but everytime I see these attempts, I just think that no non-materialism adds an iota of understanding. Worse, in fact. Whatever explains consciousness will be deeply counterintuitive. That's no strike against a theory
For any given notation, the highest integer is the one for which all available matter is consumed by it's representation. Integers aren't immaterial. They're abstractions, sure, but abstractions only exist while they're represented, using some notation on some material substrate.
(An example representation of an integer could be as simple as Arabic numerals in chalk on a board, or as complex as standing waves of connection weights in a neutral net. But it's always material, at bottom.)
If an integer is something that only has meaning within a context of notations, why is it meaningful to first reason about properties of, say, grains of sand or marks of chalk, and then claim that this reasoning also applies to other things, like, say people?
Is there something that is 'the same' about 5 apples, 5 people, 5 marks of chalk, and 5 grains of sand?
I’m trying to carefully stay inside the line of “understanding other people” instead of “trying to push my view on other people.” Do you mind explaining why you believe it’s “always material at bottom?” What does “bottom” mean here? Is there a “top” as well?
First: I appreciate how hard you work at staying inside that line! It's a rare treat to find opposing views thoughtfully and honestly pursued. (...but it also exercises MY ability to stay inside that line. If/when I fail, please point it out and I'll correct!)
Now: "Is there something that is 'the same' about 5 apples, 5 people, 5 marks of chalk, and 5 grains of sand?"
TL;DR: the thing that is 'the same' exists in our model of the world, which factors out "five-ness" as a feature all the sets have in common. That model correlates to physical brain states. No non-material explanation required.
First, notice how important units are in math - they are the link between the math level of abstraction and the underlying reality that we use math to help us model. We look at two things, then we notice they are both groups-of-five, and then we normalize our model to factor out cardinality. The number of base model concepts involved in comparing these four sets is pretty big, actually! Only some of those model concepts count as math, like the concept of a set and its cardinality. But the concepts of apple, people, chalk, and sand aren't really math, are they? And the concepts of "marks" and "grains" are kind of math - they're units, right? Kind of?
Sometimes I think about the difference between math and "the real world" in terms of [the cluster structure of thing-space](https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace) -- the cluster structure is necessary because no two apples are exactly alike. But in math, "5" is a concept, so every "5" can be exactly the same, so the cluster structure isn't necessary. I think we still use the cluster structure, but the clusters are very tight and crisp, rather than loose and fuzzy like with "real" things - and almost certainly the neuronal patterns I use to think about the "5" apples versus the "5" people are not identical. But the apparent difference in fuzziness is stark, and it makes math feel like a fundamentally different class of thing from an apple.
But the cluster-structure way of looking at it lets you consider math as both a tool you can use to model things, and a thing to be modeled itself. If you just let math be a way of modeling things, a lot of the mystery dissipates, especially if you compare it to other ways of modeling things. Like, the question "if there's no one there to do math, does it still exist?" Is like asking "if there's no one there to compress a file, does the compressed version exist?" I mean, sort of? As a model of a possible future in the mind of the person asking the question?
You can think about airplanes, and understand that at a fundamental level they don't exist (it's just quarks!) and yet be OK with the idea that an abstracted "airplane" concept is useful. Same/same with "5" and other abstracted math concepts.
You have considered the hypothesis that numbers are real material entities, and the hypothesis that they are real but immaterial entities. These do not exhaust the hypothesis space, because there is also the hypothesis that numbers are not real , that they are fictional. Clearly, we can reason about fictional entities, holding that orcs exist in Middle Earth. not Narnia and so on. If we can reason about propositions concerning fictional entities, then they are meaningful enough -- there is no reason
to suppose that extension or reference is the only kind of meaning. Fictionalism is related to formalism: formalism is the idea that maths is an invented game. So long as everyone follows the same rules, everyone can agree on mathematical "truth".
This was a pretty interesting article, though I think I disagree with you on basically every point. I'd argue that there is a highest number, sort of like how there is a fattest man. There could always conceivably be a person which is fatter, but definitively there exists a number which is the highest. This probably arises naturally from how I view math, as a tool to aid viewing the universe rather than a base tenet of the universe itself, so maybe I'm off on that, who knows.
I'm not sure that the consciousness argument is really a disproof of materialism, so much as it is a weak point. Surely we can agree that, if consciousness exists in the material realm, it must be quite complex, correct? Is it really that surprising that we struggle to understand a really complicated thing? A more concrete definition of material consciousness could arise within the next century, for all we know, as knowledge of the inner workings of the human body continues to grow. That said, I don't want to be infinitely gratuitous on this point. If materialism fails to adequately explain consciousness even with a sustained knowledge of mental processes, then you're probably on point here. I just don't think it's fair to call this a failure yet, only a work in progress.
Finally, I don't know why materialism needs an ultimate claim to truth. Its intention, best I can gauge, is to limit the scope of investigation to areas where palpable feedback can be gained. The issue with most non-materialist thought is that it lacks meaningful contradiction. There isn't a reliable way to tell if you are right or wrong about something. By tying investigations to the physical realm, materialism loses distant quantities, which rarely contradict the thinker, and replaces them with a focus on near ones, which often do. Add to this the fact that non-materialist thought also has a tendency to "lead us to a place where social consensus is mistaken for truth", but further invalidates objections to that social order, because it isn't reliant on palpable proofs. I agree that materialism is limiting, and it's certainly worth considering where those limitations might bias us away from finding Truth, but I largely see it as a constructive force.
If we consider that material objects exist in four dimensions, three of space and one of time, then I would say "there is no highest number" maps onto the infinite progession of time.
I haven't thought of an experiment. The ideas you put forth are very new to me, I'm still trying to get some traction with them.
While pondering this, I recalled that, for sets of numbers, there are different kinds of infinity. Some infinite sets, like integers and rational numbers, are countable. We can count them, but the counting never terminates. For example, there are an infinite number of rational numbers bewteen 0 and 1, but we can count them: 1/2, 1/3, 2/3, 1/4, 3/4.... The set of real numbers between 0 and 1 is also infinite, but it can't be counted.
I'm not sure if this idea of countable and uncountable infinite sets is applicable here.
What if math (and compsci, and consciousness) don't exist as such? What if they are merely ("merely") representations in neutral nets? Intermediate products on the road from sensory input to intelligent decision-making, that only exist in the model that the intelligence works with?
I don’t think there’s any way to resolve this question - what I’m trying to do is arrange my beliefs in such a way that minimizes the total number of places for “but why?” to arise. This is easier for me to do if I assume truth just arises naturally on its own, giving rise to consciousness, with the material world being a particularly interesting portion of the entire topology of truth.
I disagree! The question resolves - or rather, dissolves - when you realize that math and compsci and consciousness are representations of reality, rather than reality itself. They are real things, don't get me wrong, but they are maps, not territory. Math is a map, as material-bound as the territory it attempts to depict. (Well, to nit-pick a bit, math is a set of map-making techniques, not a map per se... but hopefully you see what I mean...)
I agree that math can be see as a representation of reality, and not real in and of itself.
But you can also do the opposite, and see math as real and physical reality as coming out of it.
You’re using the world “realize” here which means, in effect, “if you stop believing X you will see that X is not true.” Which is, of course, a tautology.
Can we experimentally determine whether one of these perspectives is correct and the other isn’t?
Math and reality are different, though. Back before Newton, flung objects fell in parabolic curves, even though no one knew the math. If our ability to do math disappears, flung objects will continue to do so. But if reality goes away, so does math. To say that math exists without reality is like saying that mapping techniques exist without any territory.
ETA: I realize I glossed over your statement that you can "see math as real and physical reality as coming out of it." I disagree with this. Math is a set of tools for describing reality. Consider: what would it take to convince you that 2 + 2 = 3 ?
Well, it's not like I know how to create some reality in a lab, so obviously I don't understand it perfectly. But basically, like, if there is nothing there to describe, there is no math to describe it.
Saying 2+2=4 is definitionally true, but it doesn’t cause 2 things, and 2 more things, to exist, in order to be 4. As far as can tell, your post is arguing that it does.
Seems to me there are many human-biased assumptions there.
Why is physics "interesting"?
Why explore something merely because it's "interesting"?
Yeah, physics is interesting to (a very small handful of) humans, because we've been programmed by evolution to seek out new things in order to improve our evolutionary fitness.
But there's nothing that makes it fundamentally "interesting".
Furthermore, consciousness (which isn't an entity) has no agency to create things to explore truth, even if "consciousness" wanted to, or were capable of wanting to.
I consider them roughly equivalent to “why are the laws of physics what they are,” which is also something I consider to be a good question. Materialists do not consider that to be a good question. The only valid materialist response to “why are the laws of physics what they are?” is “this question is meaningless.” Once you ponder the boundary conditions of the physical universe, you are leaving the materialist reservation. As someone who is curious and open minded, I consider that to be a problem with materialism.
If we reject materialism as the source of all truth and see it only as a specific filtered subset of truth, the response is “try coming up with ideas and use thought experiments to see what works and what doesn’t.”
The half baked “best answer” I have right now says that qualia exist a priori, as part of truth, like numbers or polynomials. A complex qualia called “being a self-aware organism embodied in a universe whose state evolves according to our laws of physics,” has this interesting property of allowing for a quailia called “experiencing a stream of predominantly positive-valence qualia while still enjoying the fun of being at risk if negative qualia.”
I have noticed a general pattern whereby attention seems to move from simpler to more complex games, which roughly mirrors the pattern by which energy moves from lower entropy (simpler) to higher entropy systems.
So no, I don’t have great answers here. Any theory that doesn’t leave me with either more questions, or a bunch of obviously true things that generates, eventually, the physical universe at the present moment as a likely outcome, is ultimately unsatisfactory to me.
Great article! You nailed the two major areas where a purely materialist view breaks down: math and consciousness.
Penrose has a really sophisticated understanding of how these things interweave. I highly recommend the first few chapters of Road to Reality (but the book as a whole is really dense, I had to fight my way through it). Here’s a good summary: https://www.futilitycloset.com/2019/11/12/an-eternal-triangle/
Excellent post! I broadly agree, but have a suspicion that mathematical ‘truths’ are really just tautologies that are true by definition, they just seem profound because our minds are so limited.
I also think materialism is just a shared conscious inference — I think you made this point really well.
In the end, the only thing that we’re really sure of is that consciousness exists, though. This makes me lean towards idealism
Totally agree that mathematical truths are tautologies. But that, in and of itself, is very interesting to me!
A "mathematical truth" isn't true in absentia of axioms and production rules. Once you give the axioms and reduction rules, what you have when considering a "mathematical truth" is really an asymmetric tautology, where one side is really simple (just axioms and production rules) and the other is some theorem that can be derived by operating on the axioms with the production rules. It would be wildly, wildly complex - and yet in the end, it's still 'just' a tautology.
This is what i find to be so cool about mathematical truth; you can start with nothing, pull some rules out of thin air, and then say "X,Y,Z rules lead to theorem P", and _THAT_ is now suddenly real, true, universal, regardless of what the rules of physics are wherever you happen to be.
> In the end, the only thing that we’re really sure of is that consciousness exists, though.
I think i understand this perspective, but i don't share it. I feel sure of other things, but then this makes me realize that a lot of people seem to think 'one ought not feel certain about things unless criterion x is met'. And it seems like a lot of arguments or disagreements ultimately come down to different norms about when it is, or is not, appropriate to feel certain.
This post made my day, Mark! I have suggested it as reading for my local ACX discussion group. Will have to reread and follow all the links, but so far it warms the heart of this middle-aged Catholic convert with a BS in physics at a dark time of year.
Thank you, I'm glad you enjoyed it!
Sorry, but everytime I see these attempts, I just think that no non-materialism adds an iota of understanding. Worse, in fact. Whatever explains consciousness will be deeply counterintuitive. That's no strike against a theory
Is there a highest integer? If so, what is it? Approximations are fine.
If not, what is the meaning of that fact, in material terms?
Either way, can we experimentally verify whether that belief is indeed true?
For any given notation, the highest integer is the one for which all available matter is consumed by it's representation. Integers aren't immaterial. They're abstractions, sure, but abstractions only exist while they're represented, using some notation on some material substrate.
(An example representation of an integer could be as simple as Arabic numerals in chalk on a board, or as complex as standing waves of connection weights in a neutral net. But it's always material, at bottom.)
> For any given notation, the highest integer
If an integer is something that only has meaning within a context of notations, why is it meaningful to first reason about properties of, say, grains of sand or marks of chalk, and then claim that this reasoning also applies to other things, like, say people?
Is there something that is 'the same' about 5 apples, 5 people, 5 marks of chalk, and 5 grains of sand?
I’m trying to carefully stay inside the line of “understanding other people” instead of “trying to push my view on other people.” Do you mind explaining why you believe it’s “always material at bottom?” What does “bottom” mean here? Is there a “top” as well?
First: I appreciate how hard you work at staying inside that line! It's a rare treat to find opposing views thoughtfully and honestly pursued. (...but it also exercises MY ability to stay inside that line. If/when I fail, please point it out and I'll correct!)
Now: "Is there something that is 'the same' about 5 apples, 5 people, 5 marks of chalk, and 5 grains of sand?"
TL;DR: the thing that is 'the same' exists in our model of the world, which factors out "five-ness" as a feature all the sets have in common. That model correlates to physical brain states. No non-material explanation required.
First, notice how important units are in math - they are the link between the math level of abstraction and the underlying reality that we use math to help us model. We look at two things, then we notice they are both groups-of-five, and then we normalize our model to factor out cardinality. The number of base model concepts involved in comparing these four sets is pretty big, actually! Only some of those model concepts count as math, like the concept of a set and its cardinality. But the concepts of apple, people, chalk, and sand aren't really math, are they? And the concepts of "marks" and "grains" are kind of math - they're units, right? Kind of?
Sometimes I think about the difference between math and "the real world" in terms of [the cluster structure of thing-space](https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace) -- the cluster structure is necessary because no two apples are exactly alike. But in math, "5" is a concept, so every "5" can be exactly the same, so the cluster structure isn't necessary. I think we still use the cluster structure, but the clusters are very tight and crisp, rather than loose and fuzzy like with "real" things - and almost certainly the neuronal patterns I use to think about the "5" apples versus the "5" people are not identical. But the apparent difference in fuzziness is stark, and it makes math feel like a fundamentally different class of thing from an apple.
But the cluster-structure way of looking at it lets you consider math as both a tool you can use to model things, and a thing to be modeled itself. If you just let math be a way of modeling things, a lot of the mystery dissipates, especially if you compare it to other ways of modeling things. Like, the question "if there's no one there to do math, does it still exist?" Is like asking "if there's no one there to compress a file, does the compressed version exist?" I mean, sort of? As a model of a possible future in the mind of the person asking the question?
You can think about airplanes, and understand that at a fundamental level they don't exist (it's just quarks!) and yet be OK with the idea that an abstracted "airplane" concept is useful. Same/same with "5" and other abstracted math concepts.
You have considered the hypothesis that numbers are real material entities, and the hypothesis that they are real but immaterial entities. These do not exhaust the hypothesis space, because there is also the hypothesis that numbers are not real , that they are fictional. Clearly, we can reason about fictional entities, holding that orcs exist in Middle Earth. not Narnia and so on. If we can reason about propositions concerning fictional entities, then they are meaningful enough -- there is no reason
to suppose that extension or reference is the only kind of meaning. Fictionalism is related to formalism: formalism is the idea that maths is an invented game. So long as everyone follows the same rules, everyone can agree on mathematical "truth".
This was a pretty interesting article, though I think I disagree with you on basically every point. I'd argue that there is a highest number, sort of like how there is a fattest man. There could always conceivably be a person which is fatter, but definitively there exists a number which is the highest. This probably arises naturally from how I view math, as a tool to aid viewing the universe rather than a base tenet of the universe itself, so maybe I'm off on that, who knows.
I'm not sure that the consciousness argument is really a disproof of materialism, so much as it is a weak point. Surely we can agree that, if consciousness exists in the material realm, it must be quite complex, correct? Is it really that surprising that we struggle to understand a really complicated thing? A more concrete definition of material consciousness could arise within the next century, for all we know, as knowledge of the inner workings of the human body continues to grow. That said, I don't want to be infinitely gratuitous on this point. If materialism fails to adequately explain consciousness even with a sustained knowledge of mental processes, then you're probably on point here. I just don't think it's fair to call this a failure yet, only a work in progress.
Finally, I don't know why materialism needs an ultimate claim to truth. Its intention, best I can gauge, is to limit the scope of investigation to areas where palpable feedback can be gained. The issue with most non-materialist thought is that it lacks meaningful contradiction. There isn't a reliable way to tell if you are right or wrong about something. By tying investigations to the physical realm, materialism loses distant quantities, which rarely contradict the thinker, and replaces them with a focus on near ones, which often do. Add to this the fact that non-materialist thought also has a tendency to "lead us to a place where social consensus is mistaken for truth", but further invalidates objections to that social order, because it isn't reliant on palpable proofs. I agree that materialism is limiting, and it's certainly worth considering where those limitations might bias us away from finding Truth, but I largely see it as a constructive force.
I enjoyed reading this piece.
If we consider that material objects exist in four dimensions, three of space and one of time, then I would say "there is no highest number" maps onto the infinite progession of time.
This is a good argument! It says that for every of time there will always be a future.
Is there some experiment we could use to tell if this were true?
I haven't thought of an experiment. The ideas you put forth are very new to me, I'm still trying to get some traction with them.
While pondering this, I recalled that, for sets of numbers, there are different kinds of infinity. Some infinite sets, like integers and rational numbers, are countable. We can count them, but the counting never terminates. For example, there are an infinite number of rational numbers bewteen 0 and 1, but we can count them: 1/2, 1/3, 2/3, 1/4, 3/4.... The set of real numbers between 0 and 1 is also infinite, but it can't be counted.
I'm not sure if this idea of countable and uncountable infinite sets is applicable here.
What if math (and compsci, and consciousness) don't exist as such? What if they are merely ("merely") representations in neutral nets? Intermediate products on the road from sensory input to intelligent decision-making, that only exist in the model that the intelligence works with?
I don’t think there’s any way to resolve this question - what I’m trying to do is arrange my beliefs in such a way that minimizes the total number of places for “but why?” to arise. This is easier for me to do if I assume truth just arises naturally on its own, giving rise to consciousness, with the material world being a particularly interesting portion of the entire topology of truth.
I disagree! The question resolves - or rather, dissolves - when you realize that math and compsci and consciousness are representations of reality, rather than reality itself. They are real things, don't get me wrong, but they are maps, not territory. Math is a map, as material-bound as the territory it attempts to depict. (Well, to nit-pick a bit, math is a set of map-making techniques, not a map per se... but hopefully you see what I mean...)
I agree that math can be see as a representation of reality, and not real in and of itself.
But you can also do the opposite, and see math as real and physical reality as coming out of it.
You’re using the world “realize” here which means, in effect, “if you stop believing X you will see that X is not true.” Which is, of course, a tautology.
Can we experimentally determine whether one of these perspectives is correct and the other isn’t?
Math and reality are different, though. Back before Newton, flung objects fell in parabolic curves, even though no one knew the math. If our ability to do math disappears, flung objects will continue to do so. But if reality goes away, so does math. To say that math exists without reality is like saying that mapping techniques exist without any territory.
ETA: I realize I glossed over your statement that you can "see math as real and physical reality as coming out of it." I disagree with this. Math is a set of tools for describing reality. Consider: what would it take to convince you that 2 + 2 = 3 ?
Please say more about “if reality goes away.”
What does that mean?
Well, it's not like I know how to create some reality in a lab, so obviously I don't understand it perfectly. But basically, like, if there is nothing there to describe, there is no math to describe it.
Your notion seems either circular or nonsensical.
Saying 2+2=4 is definitionally true, but it doesn’t cause 2 things, and 2 more things, to exist, in order to be 4. As far as can tell, your post is arguing that it does.
If you are asking, ”how does truth existing lead to a physical world”, I think you need consciousness in there as well.
If both exist, then consciousness explores physics because it’s an interesting aspect of truth to explore.
Seems to me there are many human-biased assumptions there.
Why is physics "interesting"?
Why explore something merely because it's "interesting"?
Yeah, physics is interesting to (a very small handful of) humans, because we've been programmed by evolution to seek out new things in order to improve our evolutionary fitness.
But there's nothing that makes it fundamentally "interesting".
Furthermore, consciousness (which isn't an entity) has no agency to create things to explore truth, even if "consciousness" wanted to, or were capable of wanting to.
These are good questions!
I consider them roughly equivalent to “why are the laws of physics what they are,” which is also something I consider to be a good question. Materialists do not consider that to be a good question. The only valid materialist response to “why are the laws of physics what they are?” is “this question is meaningless.” Once you ponder the boundary conditions of the physical universe, you are leaving the materialist reservation. As someone who is curious and open minded, I consider that to be a problem with materialism.
If we reject materialism as the source of all truth and see it only as a specific filtered subset of truth, the response is “try coming up with ideas and use thought experiments to see what works and what doesn’t.”
The half baked “best answer” I have right now says that qualia exist a priori, as part of truth, like numbers or polynomials. A complex qualia called “being a self-aware organism embodied in a universe whose state evolves according to our laws of physics,” has this interesting property of allowing for a quailia called “experiencing a stream of predominantly positive-valence qualia while still enjoying the fun of being at risk if negative qualia.”
I have noticed a general pattern whereby attention seems to move from simpler to more complex games, which roughly mirrors the pattern by which energy moves from lower entropy (simpler) to higher entropy systems.
So no, I don’t have great answers here. Any theory that doesn’t leave me with either more questions, or a bunch of obviously true things that generates, eventually, the physical universe at the present moment as a likely outcome, is ultimately unsatisfactory to me.