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>> The entire analogy of the CPU is to implement pure boolean logic through strict controls and limits of voltage levels in analog circuits. But this isn't the

>> The entire analogy of the CPU is to implement pure boolean logic through strict controls and limits of voltage levels in analog circuits. But this isn't the whole story either, right? What is "pure boolean logic" anyway? It doesn't exist even at the lowest levels. Logic is always applied to a statement of some kind, it doesn't exist as a pure abstraction. It's interesting you bring up Godel. Russell and Whitehead attempted to map logic to math in the Principia, and were refuted by Godel's incompleteness. I don't want to make too much of incompleteness (as many do), but let's not make too little of it either. At some point logic can only do so much for you, and you have to resort to analogy. I think this happens much earlier in CPU architecture than you're allowing for. What you have at a low level on the CPU are a set of instructions arbitrarily (to some degree) defined to efficiently model analogical actions like "push" and "move" and "shift" and the fundamental operations that map to assembly. And here we've already moved out of the realm of pure logic -> math envisioned by Russell and we've created logic -> language, by way of analogy. This is, I would argue, a rhetorical move rather than a logical one. We've already prepared/redacted the ones and zeroes for human consumption, and assumed a specific semantic to provide for a human imagination that will accept and adapt to the semantic.

kruhft 13y ago
The instructions in the CPU are no different from the statements in the Principia, just finite and physically realized. The instructions are combinations of logic operations, although the logic is now sequential and having state, which I would think sets it apart from the Principia since I haven't yet gotten my copy to study. The sequentiality gives rise to monadic world state, but all is still reducable to logic in the end. There's no magic in the computer and no analogy other than designing analog circuits that clamp voltage levels and a very large finite state machine. It's logic all the way down, limits and all, though instead of incompleteness there's incomputability. 2 sides of the same coin. Pure boolean logic can be reduced to symbol manipulation using rules of substituion and inferrence, but, of course, what is that composed of? Godel showed that metalogic is made of logic, you just elevate the types. What is logic made of? BTW I know computer architecture very well. There's no analogies, just abstraction.
- davesims 13y ago
  Just as a preface -- this is a great conversation, very provocative to me. If I make a misstep with regards to to CPU arch, feel free to correct (naturally). "The instructions in the CPU are no different from the statements in the Principia" As the goal of the PM was to show that math was a function of logic, we can use 'statements in the Principia' and 'the classical rules of logic' interchangeably in this context. There's two questions at play in this conversation. One is whether in principle the question of the meaning, as such, of software, can be considered apart from the whole act of perceiving the software, either as code or as the final UX. As the idea of "meaning" always involves some kind of communication, a sign or set of signs shared between two subjects, and as the idea of software pointing back to its own internal state as the content of this communication would render the communication pointless and the signs merely tautological, I think the question of meaning must bring the subject, and the act of perception and interpretation, into consideration. Scientific statements of fact, regarding the physical position of all of the machine's atoms and electrons, have content, but not "meaning" in the common sense of that term intended in the current context. The second question, which I'm pretty much winging here (although I have long-held opinions about the PM and its relationship to the same kinds of considerations about "meaning"), is whether CPU instruction sets, as subsets of the rules of logic, dictated by the fundamentals of CS (Babbage, Turing, et al), are "pure" expressions of logic, or whether in the act of abstracting logic into the restricted set of instructions, we've performed an analogy. I think the same question comes to bear -- it matters which perspective you're taking, whether "meaning" is merely a construction that hides the more essential question of physicality, or whether ideas like "push" applied to things like registers and addresses and binary values are linguistic in nature and convey more than just tautological facts about the physical state of CPU internals. I guess this pushes the question out -- why does a computer exist? To represent to the world the bare facts of its own internal state? Of course not. A 64-bit address that points to a sequence of binary values is never just that, it's always a sign that points to something else -- a person's address; a representation of an English character; a shipment of goods; a lat/long location of some specific place in the world. So, a "move" or "shift", to you, may reduce backwards to a collection of logical operations. But even there, what has happened? Nothing's "moved" or "pushed", that's just an linguistic trick to help us think of the electrical currents being turned "off" and "on" (and even those are analogies) in mentally-manageable concepts that map to the rules of logic. To me the question of "meaning" has already been broached at this point, if we wish to see computations as significant beyond, again, tautological statements about CPU internals. If we take the latter point of view, we have mere mathematical statements that have the same point as pure math -- the elegance (if that) of its own proof, not any relationship to something external to itself. But of course, that was never the real point of computers from the very beginning. The Godel/Russell/Whitehead question is apt here, because this is a very similar question, if not the same question. Is math a function of logic -- that is to say, can math be "true"? The point of the Principia was to bring about the dream of the Vienna Circle, a purely mathematical and systematic technical language to resolve all philosophical ambiguity once and for all. While there remain a few holdouts for that dream, the general consensus is that this did not work out -- Godel was only one of many problems with Positivism specifically, and I would argue, Analytical Philosophy in general, although in recent years the Analytical/Continental divide has been shortened. "What is logic made of?" Yeah. This. That's the question driving all of this. Good conversation.

kruhft 13y ago

The instructions in the CPU are no different from the statements in the Principia, just finite and physically realized. The instructions are combinations of logic operations, although the logic is now sequential and having state, which I would think sets it apart from the Principia since I haven't yet gotten my copy to study. The sequentiality gives rise to monadic world state, but all is still reducable to logic in the end. There's no magic in the computer and no analogy other than designing analog circuits that clamp voltage levels and a very large finite state machine. It's logic all the way down, limits and all, though instead of incompleteness there's incomputability. 2 sides of the same coin. Pure boolean logic can be reduced to symbol manipulation using rules of substituion and inferrence, but, of course, what is that composed of? Godel showed that metalogic is made of logic, you just elevate the types. What is logic made of? BTW I know computer architecture very well. There's no analogies, just abstraction.

davesims 13y ago
Just as a preface -- this is a great conversation, very provocative to me. If I make a misstep with regards to to CPU arch, feel free to correct (naturally). "The instructions in the CPU are no different from the statements in the Principia" As the goal of the PM was to show that math was a function of logic, we can use 'statements in the Principia' and 'the classical rules of logic' interchangeably in this context. There's two questions at play in this conversation. One is whether in principle the question of the meaning, as such, of software, can be considered apart from the whole act of perceiving the software, either as code or as the final UX. As the idea of "meaning" always involves some kind of communication, a sign or set of signs shared between two subjects, and as the idea of software pointing back to its own internal state as the content of this communication would render the communication pointless and the signs merely tautological, I think the question of meaning must bring the subject, and the act of perception and interpretation, into consideration. Scientific statements of fact, regarding the physical position of all of the machine's atoms and electrons, have content, but not "meaning" in the common sense of that term intended in the current context. The second question, which I'm pretty much winging here (although I have long-held opinions about the PM and its relationship to the same kinds of considerations about "meaning"), is whether CPU instruction sets, as subsets of the rules of logic, dictated by the fundamentals of CS (Babbage, Turing, et al), are "pure" expressions of logic, or whether in the act of abstracting logic into the restricted set of instructions, we've performed an analogy. I think the same question comes to bear -- it matters which perspective you're taking, whether "meaning" is merely a construction that hides the more essential question of physicality, or whether ideas like "push" applied to things like registers and addresses and binary values are linguistic in nature and convey more than just tautological facts about the physical state of CPU internals. I guess this pushes the question out -- why does a computer exist? To represent to the world the bare facts of its own internal state? Of course not. A 64-bit address that points to a sequence of binary values is never just that, it's always a sign that points to something else -- a person's address; a representation of an English character; a shipment of goods; a lat/long location of some specific place in the world. So, a "move" or "shift", to you, may reduce backwards to a collection of logical operations. But even there, what has happened? Nothing's "moved" or "pushed", that's just an linguistic trick to help us think of the electrical currents being turned "off" and "on" (and even those are analogies) in mentally-manageable concepts that map to the rules of logic. To me the question of "meaning" has already been broached at this point, if we wish to see computations as significant beyond, again, tautological statements about CPU internals. If we take the latter point of view, we have mere mathematical statements that have the same point as pure math -- the elegance (if that) of its own proof, not any relationship to something external to itself. But of course, that was never the real point of computers from the very beginning. The Godel/Russell/Whitehead question is apt here, because this is a very similar question, if not the same question. Is math a function of logic -- that is to say, can math be "true"? The point of the Principia was to bring about the dream of the Vienna Circle, a purely mathematical and systematic technical language to resolve all philosophical ambiguity once and for all. While there remain a few holdouts for that dream, the general consensus is that this did not work out -- Godel was only one of many problems with Positivism specifically, and I would argue, Analytical Philosophy in general, although in recent years the Analytical/Continental divide has been shortened. "What is logic made of?" Yeah. This. That's the question driving all of this. Good conversation.