'Phenomenal red' is an estimator of 'optical red' in common situations. No necessary relationship between phenomenal colour and optical qualities can be defined nor established, due to the impossibility of exhaustively specifying and testing their relationship.

First start with the notion of material-implication:

Classically, A=>B means that if A is true then B is true,and is equivalent to NOT A is True OR B is True.
Constructively, A=>B only means that a proof of B can be derived from a proof of A, and says nothing about the actual truth or provability of A or B.

The classical interpretation of material implication would say that you cannot be a lizard because you don't like flies, which shouldn't be problematic to assert, assuming that we live in a closed world containing a finite number of lizards that we can count in order to check their taste for flies.

But in the event we live in an open world containing a potentially infinite number of lizards, the classical interpretation runs into a problem in that the truth of A=>B can never be verified, implying that A=>B can never be asserted. And yet we do use conditionals without assuming that we live in finite closed worlds, which indicates our actual use of material implication is constructive rather than classical. For example, our definition as to what a lizard is includes the fact it eats flies, and therefore A=>B becomes somewhat tautologous.

Because I have not seen any resolution to these questions, I would not say that a "rule" has any existence at all. — Metaphysician Undercover

I would say that the laws of Mathematics and Logic are normative principles pertaining to conduct regulation so as to make the world easier to describe and manipulate.

These normative principles cannot be given a logical justification on pain of circularity, rather their justification stands or falls with their general overall usefulness.

In your view, what is a belief, and how should the object of a person's belief be ascertained?

Should the object of a person's belief be identified with the physical causes of their belief, in which case every belief is seen to be necessarily true when it is physically understood, or should the object of a belief be decided impersonally by linguistic convention so as to reflect the normative values of the person's community?

Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry

Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs.

In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow.

The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory.

BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games.

As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model.

Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that? — fishfry

emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model..

By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits.

Now obviously, any countable list of Provably Cauchy-Convergent Total Functions is unfinished, in the sense that a further PCCTF can be built that is is not already in the list via a diagonal argument. No problem, we just shuffle along the existing enumeration to add the new function into the existing list. But then doesn't this contradict the notion that our previous list was complete?

There seems to be an ambiguity between two definitions of completeness. If Dedekind completeness is understood to be an axiom of construction then it is trivially satisfiable in the sense that the axiom itself can be used to assist in the generation of a real from an existing list of real numbers. After all, if there wasn't a countable model of the Axioms of the reals, then they would be inconsistent, since Second-order quantification can always be interpreted as referring only to the sets constructively definable in first-order logic.

On the other hand, if completeness is understood to refer to a finished list of PCCTFs, our list is not complete in that sense.

So it seems to me that countable model of reals, both first and second order, are especially useful ( not to mention the only models we use in practice),for clarifying the relationship between Dedekind completion, Cantor's theorem and ordered fields.

If one abandon's the second-order completeness axiom, and possibly cauchy convergence, then there are less constraints in the construction process, allowing one to define a potentially larger field of computable numbers that includes infinitesimals as is done with the (constructive) Hyperreals, and one can even include computable 'numbers' that are aren't provably total. In which case ones countable list is now finished, but now there are no more numbers to be added, because now the diagonal argument cannot be used to construct a new numbers in virtue of one's list including non-numbers that aren't guaranteed to halt on their inputs.

So i hope this had lead to a satisfactory conclusion.

The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D. — fishfry

?? Perhaps I should have been clearer from the beginning, but i took everyone's understanding for granted that a computable number refers (in some way) to a computable total function. Apologies if that is the case. For surely you appreciate that the computable total functions aren't countable?

The computable total functions are a proper subset of the computable functions that also contain partial functions. i.e. that do not halt on a given input.

It is true to say that the whole set of computable functions is countable, for reasons you'e sketched. It is not true to say that the set of computable total functions are countable, for we cannot solve the halting problem. Hence the reason why we say the computable numbers are sub-countable: the only way we could 'effectively' enumerate the computable numbers is to simulate every Turing machine and wait forever, meaning that any 'candidate enumeration' we construct of our computable numbers after waiting a finite time is also going to contain computable functions that aren't total and hence are not numbers.

For the constructivist, this "subcountability" is all 'that 'uncountability' means. It is simply means that we can never construct a total surjective function from the natural numbers onto the computable numbers. It doesn't mean in any literal sense that we have more computable real numbers than natural numbers.

The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist. — fishfry

We have to be careful there. We can run every Turing Machine and at any given time create a bar-chart of the ones which have halted, and this histogram comprises a sequence of computable functions whose limit isn't a computable function. To my understanding this sequence of functions isn't cauchy convergent, for we cannot construct a bound on the distance between successive histograms. Let's not forget that there are an infinite number of computer programs of every size.

Compare this situation to a computable total function f(n) representing the "values" of the Goldbach's Conjecture; Let's say that f(n) = 0 if every even number less than n is the sum of two primes, otherwise f(n)=1. Here we can also compute the individual digits in finite time. If GC is decidable, i.e. GC OR ~GC, then f(n) is Cauchy convergent to either 0 or 1. But if GC isn't decidable, then as with Chaitin's constant f(n) doesn't have a cauchy convergent limit, even though f(n) is a computable total function.

Therefore, in order to know that one has constructed a complete and ordered field of computable numbers, one must only use a set of provably Cauchy-convergent computable total functions, for which every cauchy-convergent sequence of these functions is also provably cauchy-convergent.

Too few, clearly. There are only countably many of them.
...
And no countable ordered field can be complete. It's a theorem. — fishfry

The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle.

The constructive reals aren't complete because there are too few of them, only countably many — fishfry

Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions.

After all this discussion, I'm starting to reject my claim that Zeno's paradox can be solved by our inability to count and measure things. I'll think about this more at a later time. — Michael Lee

Zeno's paradox is best solved by observing how you would practically explain the paradox. To practically demonstrate the paradox requires one to repeatedly move an object along the same path, but ending the motion at the half-way point of the previously travelled distance and exclaiming "the object must have earlier travelled through this point".

In other words, a demonstration of Zeno's paradox can only explain what an object position is by destroying the object's motion. In other words, this demonstration shows that the construction of a position is incompatible with the construction of a motion, and hence is an intuitive demonstration of the Heisenberg Uncertainty Principle.

In my opinion, Zeno was close to discovering this principle characteristic of Quantum Mechanics, purely from ordinary phenomenological arguments.

Well in my constructive understanding:

The 'Second-order' reals (as described via second-order logic) are also 'unique' from a constructionist perspective; for if the Axiom of Choice is rejected then second-order quantification over the sets of reals is strictly interpreted as quantifying over the constructable-sets of reals. Consequently, what we then have is a first-order countable model of the reals in 'second order' disguise. The reason why the real number field is unique in this interpretation is because we are actually still working within first-order logic; and since the Ultrafilter Lemma isn't constructively acceptable, the Löwenheim–Skolem theorem for first order-logic that depends upon it fails. Therefore constructive first-order models of the reals only possess models of countable cardinality. Consequently, there cannot exist models of constructive reals that are "non-standard" thanks to Tennenbaum's theorem that denies the existence of non-standard countable models that are recursive.

From this constructive perspective , the semantic intuition behind CH is trivially correct: There are no subsets of R whose size is greater than N but less than R, simply because the real numbers are encodings of natural number elements (via Godel numbering of the underlying computable total functions) and therefore they are of the same number. But alas there only exists an effective algorithm for deciding the provably total functions, i.e the provable real numbers, and hence there is no constructive proof that the number of provably constructive real numbers equals the number of constructable real numbers.

What about art semi-automatically generated by a neural network that in effect produces novel images with high artistic potential by interpolating the patterns that exist within large databases of artistic, natural and cultural images? Or that transfers the statistical qualities of an artist's style onto an arbitrary image to produce a novel 'painting' in that artist's style?

Who is the artist here, and who owns the results?

In computer programs infinity is used as a free-variable that might later become bound to a random finite number. So it is a logical concept rather than a concrete number.

Can you show that idea with practical example? — Zelebg

Dark Energy hypotheses in physics are currently the most fashionable example as to why falsification isn't used in practice. Rather than considering the Hubble data of the speed of receding galaxies as refuting General Relativity, Physicists instead 'fix' GR as being true by proposing new and (individually) untestable auxiliary hypotheses so that GR still 'works in combination. In fact, to my understanding Dark energy isn't even at the stage of being a well-defined 'hypothesis'.

Auxiliary information also includes the trivial and taken-for-granted assumptions that your instrumentation is in 'full working order', that the laws of physics haven't changed since you began the experiment, that you aren't hallucinating, etc. etc. In short, no hypothesis is ever tested in isolation, and the auxiliary assumptions upon which the credibility of experiments rests aren't even exhaustively stateable, let alone formally stateable. Hence the reason why falsification isn't a good model of science or epistemic judgements in general. It's rooted in the archaic notion of logical Atomism - the idea that language has legible denotational semantics where the truth of a proposition stands or falls in isolation of the truth of every other proposition. But this is only true in toy-world scenarios described in an artificial language.

It's a complex, ill-posed and frankly outdated assertion. Firstly, an observation O can only materially entail the contradiction of a hypothesis H in a closed finite world. For in an open-world, the meaning of the material implication O => ~H isn't empirically reducible to observations, and is instead an auxiliary hypothesis, A, which isn't itself entailed by some other observation on pain of infinite regress. So in an open world we have A => ( O => ~H) , and hence O => (~A OR ~H)

- We know that we cannot enumerate all halting Turing machines, so for every supposedly complete list of halting Turing machines I can find another halting Turing machine that is not in the list.
Does that mean that the set of halting Turing machines in uncountable? No! It only means that there is no way to enumerate that list!. — Mephist

Yet saying that there is no way to enumerate the total computable functions is somewhat ambiguous, for as previously mentioned we can use brute force to simulate every TM on every input and enumerate on-the-fly the algorithms that have so far halted on all their inputs. Furthermore, the classical logician with realist intuitions will go further to argue that there is a definite matter-of-fact as to the set of computable functions and will therefore believe in the independent existence of a 'finished' enumeration, interpreting the limitations of finite constructive arguments to produce such an enumeration as being epistemic limitations rather than metaphysical limitations. Instead they will simply appeal to the Axiom of Choice to claim the independent existence of completed enumerations of the computable total functions.

As we said earlier, this full enumeration cannot be used if the diagonal function d(x) is to be both computable and total; otherwise if the realist diagonalizes the hypothetical full enumeration of computable total functions, then d(x) cannot be computable, for the enumeration ensures that d(x) is total. That d isn't computable is obvious, since it involves running nearly every Turing Machine for an infinite amount of time and then diagonalizing, meaning that it's godel number g is infinitely long and that it assumes an infinite value when evaluated at d(g)=d(g)+1

We know that Cantor's Theorem concerning the cardinality of the power-set of integers isn't a constructive proof, for we cannot enumerate and diagonalise only the Turing machines representing the recursively enumerable sets due to the Halting Problem, so we must enumerate the larger set of TMs.

And yet, the entire set of countable TMs can be diagonalised to prove that the set of countable TMs are "uncountable", by dynamically enumerating the halting TMs so as to ensure the termination of the diagonal TM for each of it's inputs; but in fact all that my proof of "uncountability" amounts to with respect to Turing Machines, is the construction of an enumeration of Turing Machines in such a way that the diagonal Turing machine cannot be part of the enumeration. This is analogous to enumerating the odd numbers and then diagonalising them to construct an even number.

To spell out the difference, in the case of Cantor's Theorem the constructed enumeration of the sets of natural numbers is considered to be prior to the construction of the diagonal set, but in the case of my method, the enumeration of TMs was constructed via the construction of the diagonal function. In other words, selectively constructing a non-exhaustive infinite enumeration via a diagonal procedure isn't a proof that a bijection with the natural numbers doesn't exist under a different enumeration. And in the case of Turing Machines we know that such a bijection does exist.

But this raises doubts about Cantor's original diagonal argument, for I might have been lucky enough with my original enumeration of TMs to produce a diagonal function without requiring any shuffling of the enumeration. Therefore Cantor's original argument isn't proof enough that the power-set of N is literally larger than N.

A proof as to why the real numbers are absolutely countable, in spite of all pretences to the contrary:

1. Enumerate the undecidable set of total functions within the entire set of enumerable Turing Machines of one argument {f1(x),f2(x),f3(x),..}, by running every Turing Machine in parallel on each input x=1,x=2,..., and shuffling their enumeration over time as necessary, so as to ensure that fn is defined when run on input x=n.

2) Define the Turing-Computable total function g(n) =fn(n)+1.

Congratulations, you've "proved" that the countable set of Turing machines is "larger" than the countable set of Turing machines.

Firstly there is the problem of confabulation, especially when subjects are pressed for more detail; are people accurately reporting their earlier experience, or are they improvising novel content over a vague memory? With memories, it's never clear which parts are authentic.

Also, what are the necessary descriptive features of an apparition? Or is an apparition more of a hypothesis about experience?

My dozen or so successful experiments with day-time and wake-induced sleep-paralysis were personally interesting, but i have no idea how to report most of them, and they were bizarre and my memory is vague. None of them involved aliens or apparitions. My favourite experience wasn't even visual, but more of a blind ecstatic trance accompanied with a feeling of moving at lightening speed and a wooshing sound like a jet engine. Ironically, the dullest experience was a high-definition visual OBE that lasted for about a minute, that felt so real and ordinary that I briefly wondered whether I was actually awake and physically in my kitchen, but then realised I must have been asleep because the kitchen clock told the wrong time, plus there was this dizzy sensation from feeling pulled around on that mythical elastic cord that I never actually saw...

An important question is the relationship of the Axiom of Choice (AC) to the Law of Excluded Middle (LEM), for Classical Logic is normally distinguished from Intuitionistic Logic on the basis of the latter axiom rather than the former axiom. Furthermore, intuitionists often claim that AC is constructively acceptable by interpreting AC to refer to the very construction of a function, for intuitionists do not accept the existence of non-constructive functions. This is very confusing, because AC's natural role is to refer to an unspecified function for which we do not possess a constructive description. This situation arises all the time in computing when a program points to an externally provided input data-stream that the programmer cannot further describe.

Now according to the SEP's article on the Axiom of Choice , AC implies LEM in the presence of two further axioms, namely Predicative Comprehension (PC) and Extensionality of Functions (EF). The former says that the image of every predicate applied to individuals is a set, whereas the latter says that every extensionally equivalent pair of sets has the same image under every Set Function.

https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog

The author of the article proves that PC & EF => ( AC => LEM), but he then argues that whereas PC is constructively valid, EF isn't. See the argument, but I don't find his argument about EF persuasive, at least as I understand it.

In my current view, EF is also constructively admissable, implying that the precise difference between classical logic and intuitionistic logic is AC as much as it is LEM, which then cements the view that classical logic describes a game between two players, whereas intuitionisitic logic describes solitaire.

A further motive for my view (and indeed the most natural motive), is that classical logic involves sequents of the form (a AND b) => x OR y, where it isn't known which of x or y is true, in which the negation of one implies the existence of the other. On the other hand intuitionistic logic only involves sequents with a single conclusion, of the form (a AND b) => x. Thus there is indeterminism in the case of classical reasoning, but not in the case of intuitionistic reasoning.

I interpret the Axiom of choice to be a 'prayer to nature' to send me the desired object already made. The axiom refers to my opponents choices that are not modelled in the formalism i am using.

If I have no internal strategy for constructing a basis from my formalism of a vector space, then I am reliant upon nature sending me a basis, which I have no control over. But suppose nature never sends me a basis?

Arguments between constructivists and classical logicians are caused by a fundamental disagreement about the nature of proof. The former equates proofs with fully-determined algorithms under the control of the mathematician, whereas the latter allows proofs to interact with nature in an empirically-contigent and indeterminate fashion.

Unfortunately, classical logicians are usually in denial about what they are actually doing. Instead of admitting that their notion of proof is empirically contigent and not internal, they insist their notion of proof is internally constructive in a transcendental platonic realm.

A non-computable real number r refers to a truly random infinite process, and yet the distinction between a truly random infinite process and a pseudo-random infinite process isn't finitely testable, since any finite prefix of r is computable. Since r cannot be finished, at any given time r can be equally interpreted as referring to an under-determined pseudo-random process. Yet any process we specify ourselves is fully determined. Therefore r can only be interpreted as referring to a process of nature that we are observing but that we ourselves haven't specified and have only incomplete knowledge of and control over.

Therefore when a physicist makes the observation x = 0.14 +/- 0.0001, he could be equally described as stating an interval of rational numbers or as stating an interval of real-numbers. If this sounds wrong, "because the real numbers are uncountable, whereas the rational numbers are countable", recall Skolem's Paradox that the set of real-numbers actually possesses a model in which they are countable. The only important thing to know is whether the physicist fixed the result or whether he measured the result, for constructing a certain number is different to measuring an uncertain number - this difference isn't easy to express in either classical or constructive mathematics.

Unless 'Living within a simulation' is defined in such a way as to be falsifiable, the hypothesis is meaningless. And yet it is imaginable that the falsification of any particular simulation hypothesis can be simulated under an alternative simulation hypothesis. Therefore it doesn't make sense to ask the question in a general absolute sense.

As a matter of fact, Wittgenstein plays an important part in my epistemology. Consider where I talk about the many uses of the word know, which is taken from the PI and especially OC. — Sam26

But what about ontics? If the meaning of a proper noun, say "Elvis Presley", is considered to be a family-resemblance of uses across partially over-lapping language-games, then the question as to whether Elvis Presley is dead or alive doesn't have a single definite answer. In Karoake, as far as i'm concerned, if it looks like Elvis and sounds like Elvis, then it's Elvis.

Our understanding of the "soul" of a person has much in common with our attribution of sense to their name, which is retained after the expiration of a referent bearing that name. If an anti-realist concerning other-minds ontologically prioritises sense over reference, he will answer questions concerning immortality very differently to the realist, for these opposing view-points use different and incompatible criteria for personal identity.

Realists also vehemently disagree with one another. One argues that "Elvis" is token-identical with the current state of a corpse buried at Graceland and that evidence for Elvis's reincarnation on earth or in heaven is therefore logically impossible. The other says "Elvis" is type-identical with a class of potential physical objects, and that reincarnation is therefore physically possible but unlikely. The presentist argues "Elvis" is meaningless because nothing bears the name in his present vicinity etc.

When you argue for the possibility of evidence of consciousness surviving the body, what is your understanding of a proper-name?

A number is a definite unit of measurement, a definite quantity. The so-called "irrational number" is deficient in the criteria of definiteness. The idea of an indefinite number is incoherent, irrational and contradictory. How would an indefinite number work, it might be 2, 3; 4, or something around there? — Metaphysician Undercover

In the example of sqrt(2), Alice creates a computer program f(n) for calculating sqrt(2) to n decimal places and sends f(n) to Bob. Bob receives f(n), then he mysteriously decides upon a value for n, then runs f(n) and sends the result to Alice.

When irrationals are identified as being rational-number generating algorithms, irrationals are obviously well-defined. But of course we are rarely interested in algorithms per-se, and we also like to use irrationals as numerals, when for instance we use them to label the length of the diagonal of a square.

Now obviously, our visual impression of a square is vague to a certain extent, as is any physical measurement of a square, and if we repeatedly measured the diagonal of an actual square with very high precision we would obtain a range of (rational valued) measurements. In other words, by labelling the diagonal of a square with, say, sqrt(2), we are using Sqrt(2) not as denoting an algorithm, but as denoting an arbitrary rational or distribution of rationals, that is yet to be determined through a process of (repeated) measurement.

This is another example of why game-theory is a good model of mathematics use.

IMO, the heart of the problem is that the notations of both classical and constructive logic do not explicitly demark the analytic or a priori uses of logic pertaining to activities of computable construction in which every sign is used to refer to a definite entity or process, from logic's a posteriori or empirically-contigent uses in which some or all of it's signs are not used to denote anything specific a priori but whose meaning is empirically contigent upon nature providing some (possibly non-existent) outcome at a future time.

This is why I consider communication games to be the most important interpretation of logic. For it identifies the constructive content of logic with the permissible sequences of actions that can be taken by player A, and the 'non-constructive' content of logic with message-replies that A receives from a player B as a result of A messaging B. The existence of a message-reply from B is not-guaranteed a priori, and A's message to B is only said to be meaningful as and when A receives a reply from B.

A constructive real number refers to an algorithm constructed by A for generating a sequence of integers. In the case of a non-constructive real number, A invokes the "Axiom of Choice" , which is interpreted as A 'outsourcing' the creation of the real number, by sending B a message requesting B to send A an arbitrary sequence of integers. The sign for a non-constructive real number has no specific meaning or referent until A receives a stream of integers from B.

Pure and unapplied logic makes no empirical claims and only expresses linguistic rules for rewriting a term. All logically valid sentences are reducible via rules of term-rewriting to "X=X", where the equality sign designates the term on the left hand side to be an abbreviation of the term on the right-hand side.

If in the course of reducing a logical expression one obtained the opposite, all this would mean is that one's use of terminology isn't consistent. The solution is either a wholesale adjustment of the axioms that defines one's terminology, or to forbid on a case-by-base basis any derivation that leads to contradiction. For example, if Peano arithmetic was discovered to be accidentally inconsistent, a possible solution is to retain it's rewriting axioms for arithmetic, but to forbid any derivation beyond a certain size.

Philosophers have an unfortunate tendency to mistake ordinary uses of equality as denoting a physical relation between things rather than as being a linguistic relation between terms. For example, if the word "Now" is considered to refer to presently moving objects we arrive at the Hegelian contradiction "Now isn't now". But all this means is that our definition of 'now' is inconsistent. The contradiction is removed by replacing each and every use of "now" with a unique and new term, such that we are never tempted into equivocating one "now" with another.

In practice, a "non-computable" process, that is to say a truly random process, is indistinguishable from an unknown pseudo-random process, since we can only ever observe a finite number of observations generated by an unknown process. For the unknown process to be declared as 'truly random' it can never terminate and it must pass an infinite number of tests that refute each and every conceivable Turing computable algorithm that could have generated it.


The practical impossibility of distinguishing an unknown pseudo-random process from a really random process provides us with ammunition for rejecting an absolute ontological distinction between randomness and lawfulness. In which case, the question as to whether nature is computable or not is meaningless.

Instead we need only invoke a game-theoretic distinction between a controlled process referring to a process we create and control ourselves using an algorithm, versus an uncontrolled process that is part of nature that we only query and make speculative computer models of.

The stance that 'knowledge consists of instructions and the ability to follow them' is the epistemological philosophy known as constructivism. In my view this is misguided, because

i) Most of our knowledge and inferences consist of analogies and analogy-making, rather than consisting of recipes and the ability to follow them. Indeed, the Church-Turing thesis is purely the expression of an analogy between mathematical formalism and practical rule-following by humans.

ii) Constructivism cannot be self-justifying without pain of infinite regress; so-called 'constructive' reasoning actually consists of implicit analogical inferences expressed as axioms that lack further constructive justification or explication.

iii) In practice most of our so-called 'constructive' inferences are outsourced to external oracles we call 'calculating devices'. But unless one holds strongly realist beliefs in causality and identifies the logical description of a machine as an expression of a physical hypothesis, the output of a calculator cannot be said to be 'constructed' from it's inputs. Indeed, a central function of logic is to be able to describe the world in an agnostic fashion without committing to speculative physical theories.

Classical Logic together with Model-Theory and the Axiom of Choice accommodates our analogical leaps of faith known as "truth by correspondence" that stem from our non-deterministic interactions with nature better than intuitionistic logic, since the latter is purely the expression of games of solitaire played according to known rules.

Do you understand Turing's answer to the Halting problem? Just as Cantor's diagonal argument shows that not every infinite set of numbers can be put into 1-to-1 correspondence with the Natural numbers, so do the various undecidability results, starting from Church-Turing thesis, show that indeed there are mathematical objects that cannot computed. Not everything can be calculated/computed by a Turing Machine. — ssu

Recall that the proofs of Godel and Cantor correspond to Turing-computable algorithms. Are you meaning to suggest that a deterministic machine can follow a set of rules to prove that there exists a theorem that cannot be deterministically derived by those rules? Or that proves in a finite number of steps the existence of a literally uncountable universe? That's what the common (mis)interpretation of Godel's and Cantor's results amounts to.

It is certainly possible that nature is really random in the sense of falsifying any proposed theory-of-everything. But this is to speculate about nature, rather than to deduce a logical conclusion.

We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided.

I personally think that absolutely everything is mathematical or can be described mathematically. Huge part is just non-computable. When we would understand just what is non-computable, we would avoid banging our heads into the wall with assuming that everything would be computable. — ssu

Computability isn't a mathematical assumption, rather computability refers to the very activity of construction by following rules. Since mathematical logic consists only of the constructive activity of rule-following, the idea that mathematical logic can capture the non-constructive notion of "non-computability" is a contradiction in terms. None of Cantor's conclusions are really captured in his (ironically constructive) syntactical expressions. He is merely projecting his theological intuitions onto logical syntax.

Science on the other hand, attempts to predict the course of nature using a particular set of rules. Yet there is no reason to believe that the course of nature follows any particular set of rules. Hence we could say that nature might be "non-computable", but this "non-computability" cannot become the object of mathematical study on pain of contradiction.

A topologically structured set that only possesses the notion of "neighbouring" elements does not in itself possess 'holes', regardless of the finiteness of the set.

The notion of a continuum is relative to the notion of a 'hole', which is describable in terms of the absence of a topology-preserving surjection between two topologically structured sets.


"Gaps" only exist within the line of countably computable reals for platonists who think that non-computable numbers exist. Even if they are granted the existence of an uncountable number of non-computable reals to occupy those "gaps", the resulting model of the reals cannot control their cardinality, suggesting to the Platonist further holes, forcing him into constructing hyperreals and so on, without ever being able to fill the gaps in the resulting continuum.

This is another good reason for rejecting the non-constructive parts of mathematical logic.

Like the morning and evening star, water and H2O, temperature and molecular motion, Samuel Clemens and Mark Twain, the empty set and 0, or the charge of every electron in the universe.

Or that damned ship that had all its parts replaced during its voyage. — Marchesk

Well obviously in each case, the terms on the left and right side do not possess the same sense, so are not practically substitutable, so are only substitutable in a theoretical sense that makes potentially falsifiable counterfactual claims. Now standard dogma alleges that Superman is only de dicto different from Clarke Kent, but that de re they are one and the same person.

" In the context of thought, the distinction helps us explain how people can hold seemingly self-contradictory beliefs.[4] Say Lois Lane believes Clark Kent is weaker than Superman. Since Clark Kent is Superman, taken de re, Lois's belief is untenable; the names 'Clark Kent' and 'Superman' pick out an individual in the world, and a person (or super-person) cannot be stronger than himself. Understood de dicto, however, this may be a perfectly reasonable belief, since Lois is not aware that Clark and Superman are one and the same. " - https://en.wikipedia.org/wiki/De_dicto_and_de_re

Yet in "Superman 3" Superman was poisoned and divided himself into Evil Superman and Clarke Kent who then fought a physical fight. I submit this as evidence that Superman is not Clarke Kent de re .

Your intuition is constructively valid, since constructivists identify the real reals with the computable binary sequences, that is to say, the binary sequences that are computable total functions, and since they reject the premise of Cantor's theorem that the power-set of the natural numbers exists (due to most of it's elements being non-computable), they instead regard diagonal arguments as merely showing that a hypothetical enumeration of only the genuine real numbers cannot be formulated.

Unfortunately in Cantors day there wasn't much attention paid to the underlying algorithms used to generate binary sequences, and hence he failed to admit the critical distinctions between

-The Total functions corresponding to the countable set of reals that can actually be constructed by an algorithm that halts to produce a digit for every argument representing a position of the sequence.

-The Provably Total functions corresponding to the subset of constructive reals that are 'a priori' deducible, in the sense that their underlying algorithms can be proved to halt without requiring the algorithms in question to be actually run.

-The Non-Provably Total functions corresponding to the subset of 'a posteriori' deducible constructable reals , whose algorithms in fact halt when run, but whose halting cannot be proved without running the algorithms in question.

-The Partial functions that fail to produce digits for every position of the sequence, and hence fail to represent a legitimate number, a property which is generally unknowable.

Cantor's diagonal argument when applied to the countable set of provably total functions, constructs a 'diagonal' total function, i.e. an additional valid real number, that is no longer provably total, and we also know that the combined set of provably total and non-provably total functions is countable (via enumerating their respective Turing machines).
Yet we cannot apply a diagonal argument to the set of non-provably total functions to produce an additional real number, for their very non-provability forbids us from knowing apriori whether or not our list contains only total functions representing genuine real numbers. Therefore there is no constructively acceptable diagonal method for producing a new real number from an enumeration of all and only the real numbers.

Identicality isn't a description of appearances, it is an adopted convention that grants the inter-substitution of two or more distinguishable things in every situation. As an adopted convention, it doesn't make sense to ask whether two things really are identical.

To me the past is a deducible concept without referencing external realities — Devans99

This expression sounds very anti-realist to my ears. Namely that the past is deducible i.e. in some sense a living construction out of present sense-data and current activities, as opposed to being an inductively inferred and immutable hidden reality that cannot be observed.

The conceptual distinction between the inductive inference of causes from effects versus the deductive 'construction' of effects given causes seems to lie at the heart of disagreements between the realist and anti-realist. The latter wants to bring these two concepts much closer together by interpreting causal induction constructively as a generalised form of deduction.

- I have thoughts, these thoughts from a causal chain. The present exists, there are thoughts that I am no longer having, so the past exists. There are thoughts that I will be having so the future exists. I can label each thought with an integer. Assuming a past eternity, then the number of thoughts would be equal to the highest number. But there is no highest number, so a past eternity is impossible? — Devans99

Yes, the realist thinks of "past eternity" metaphorically in geometric terms, as an infinitely long line beginning at, say, zero and ending at positive infinity at the "the present", which obviously cannot be constructed. The anti-realist can reverse this metaphor by labelling the present with zero, and considering the past to be 'created on the fly' as and when evidence of the past becomes available.

And of course, you presumably mean that you have present thoughts which you interpret as being 'past-indicating' and present thoughts you interpret as being 'future-indicating'. Although recall that an appearance per-se does not refer to either the past or future, as exemplified by a randomly generated image that by coincidence looks historical. And recall that we can doubt the veracity of our memories if we judge our present circumstances to contradict them. So we cannot make an immediate identification of appearances, memories, thoughts or numbers with points on a physical-history timeline.

A believe in infinite past time is therefore akin to a belief it is possible to count 'all the numbers'. — Devans99

That only holds if one is a realist about the past who believes that the object of history is an unknowable reality in being unobservable and transcending present and future information.

In contrast, according to anti-realism the very truth of a past-contingent proposition reduces to present information. if the present state of information is ambiguous with respect to two historical possibilities, then according to anti-realism there is no matter of fact as to which historical possibility is true. Moreover, since the anti-realist never wants to claim 'perfect' knowledge of the past, he must insist that the past is infinitely extensible in a literal sense as and when new information becomes available.

For example, suppose Archaeologist A at time T unearths evidence E implying the existence of a fact F at time T', where T' < T. In stark contrast to the realist, the anti-realist considers A and E to constitute part of the very meaning of F, such that the truth of F is a function of T.

There is a great deal of empirical evidence that the speed of light is a universal constant obeyed by everything in the universe; we have been measuring it for 100s of years and we currently know it within a measurement uncertainty of 4 parts per billion.

Saying the statement "nothing travels faster than light" is about the language of physics seems to me to be equivalent to saying the statement is a natural law — Devans99

Sort of, yes.

Consider the statement "All objects have a temperature at or above zero Kelvin". Interpreted from a realist's perspective, the sentence is synthetic and makes potentially falsifiable empirical claims, and in this sense is considered to be a "natural law".

But from the anti-realist's perspective, the statement is analytic and merely states that negative numbers play no role in our physical concept of temperature states; From this perspective, the statement is part of the convention of our physics language and in being fixed by convention isn't considered to be 'up for grabs'.

Yet as Quine pointed out, conventions often undergo dramatic revision every so often in order for a language to improve it's expression of new observations. But as Carnap pointed out, given any state of evidence, there is freedom as to the physics convention used. So there will invariably be disagreement as to whether any statement taken in isolation is false, true or meaningless. So in this sense there is indeed 'equivalence'.

The language of physics is our model of natural laws after all - so I maintain a belief that the natural laws of the universe are time-aware. .This suggests time is more than just a human invention. — Devans99

Unfortunately there isn't a single viable language of physics. Holistically, all viable languages account for the same observed phenomena, but each language suggests different prescriptions as to what are the most informative future experiments to conduct.