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Another statement of the Church-Turing thesis is to say that the the functions computable by a Turing machine (or lambda-computable, or general recursive) makes

Another statement of the Church-Turing thesis is to say that the the functions computable by a Turing machine (or lambda-computable, or general recursive) makes a good proxy for the intuitive concept of computability. From this perspective, "thesis" is a misnomer: it just happens to be a useful definition for "computability". That's not to say that there aren't edge cases where intuitive "computability" and Turing computability disagree; for most people, any number of oracle machine constructions will fall under the former but not the latter. We've just chosen to define the term in a way that's "easy" to define somewhat rigorously and has useful properties. It's kind of just like if we were to call the usual definition of the reals "Cauchy's thesis" or somesuch.

30f0fn 6y ago
The Church-Turing proposition carries an implicit challenge: find an algorithm to evaluate a function which is not Turing computable. Because it can be challenged in this way, it’s not just a definition (mere stipulation how to use the word “computable”).
- Konohamaru 6y ago
  Back in 200 B.C. we had something called the "Euclid-Archimedes Thesis" (okay it wasn't literally called this) which stated that every measurable curve was either a line or a circle. The idea that there was a way to measure the length of any irregular curve was simply impossible. And this thesis went unchallenged for an amazing 1,800 years until the 17th century when several mathematicians constructed measures for various irregular curves. Be very skeptical about "theses" that are handwavy about a frontier subject (pure, abstract geometry was just as much of a frontier subject as abstract machines) suspiciously lacking any proof.
- naniwaduni 6y ago
  It's pretty easy to come up with "algorithms" in the intuitive sense that are not Turing computable; just violate any of the finiteness constraints. The fact that you might object to considering such procedures algorithms rests on the fact that the "thesis" is providing a definition.
  - simcop2387 6y ago
    Turing machines and being turing computable doesn't require any finite constraints. Both infinite time and infinite storage space are expected for a universal turing machine. We obviously have never built one.
    - supercasio 6y ago
      There are many finite constraints. First, there is no infinite time [1]. The most famous uncomputable problem is the halting problem (given an algorithm A and an input x, can we compute it? that is, will A stop on x?). Although we have an infinite tape, since the time of a computation is finite, the space it uses is also. Second, we describe a Turing machine using finite sets. [1] Actually, as described by Turing, a Turing machine neves stops, but this is only because he is interested in computing real numbers (for example pi). But even using an original Turing machine, we do not "compute" pi. We only "compute" pi with some precision.
      - dfischer 6y ago
        In a way, anything discrete is a logical assertion that exists purely in imagination. Base reality is continuous in every way. It’s imaginary human reality to separate things apart from the whole. Physics and philosophy have struggled with this over atom vs non atom view of “base reality” - interesting to see how it shows itself in many different ways. Anything quantized runs into issues with consistency, as the quantity only works in an imaginary model. No model is sufficient as it has a fixed size of requirements to be a model.
        drdeca 6y ago
        I think the jury is still out on whether space is quantized. Maybe wave function values can’t be quantized, but that’s not “in every way”. And I’m pretty sure the different eigenstates for spin operator in a given direction are distinct and not a continuum?
        6y ago

30f0fn 6y ago

The Church-Turing proposition carries an implicit challenge: find an algorithm to evaluate a function which is not Turing computable. Because it can be challenged in this way, it’s not just a definition (mere stipulation how to use the word “computable”).

Konohamaru 6y ago
Back in 200 B.C. we had something called the "Euclid-Archimedes Thesis" (okay it wasn't literally called this) which stated that every measurable curve was either a line or a circle. The idea that there was a way to measure the length of any irregular curve was simply impossible. And this thesis went unchallenged for an amazing 1,800 years until the 17th century when several mathematicians constructed measures for various irregular curves. Be very skeptical about "theses" that are handwavy about a frontier subject (pure, abstract geometry was just as much of a frontier subject as abstract machines) suspiciously lacking any proof.
naniwaduni 6y ago
It's pretty easy to come up with "algorithms" in the intuitive sense that are not Turing computable; just violate any of the finiteness constraints. The fact that you might object to considering such procedures algorithms rests on the fact that the "thesis" is providing a definition.
- simcop2387 6y ago
  Turing machines and being turing computable doesn't require any finite constraints. Both infinite time and infinite storage space are expected for a universal turing machine. We obviously have never built one.
  - supercasio 6y ago
    There are many finite constraints. First, there is no infinite time [1]. The most famous uncomputable problem is the halting problem (given an algorithm A and an input x, can we compute it? that is, will A stop on x?). Although we have an infinite tape, since the time of a computation is finite, the space it uses is also. Second, we describe a Turing machine using finite sets. [1] Actually, as described by Turing, a Turing machine neves stops, but this is only because he is interested in computing real numbers (for example pi). But even using an original Turing machine, we do not "compute" pi. We only "compute" pi with some precision.
    - dfischer 6y ago
      In a way, anything discrete is a logical assertion that exists purely in imagination. Base reality is continuous in every way. It’s imaginary human reality to separate things apart from the whole. Physics and philosophy have struggled with this over atom vs non atom view of “base reality” - interesting to see how it shows itself in many different ways. Anything quantized runs into issues with consistency, as the quantity only works in an imaginary model. No model is sufficient as it has a fixed size of requirements to be a model.
      - drdeca 6y ago
        I think the jury is still out on whether space is quantized. Maybe wave function values can’t be quantized, but that’s not “in every way”. And I’m pretty sure the different eigenstates for spin operator in a given direction are distinct and not a continuum?
        6y ago