Geez, is this really the best OCR we have in 2022? This is the text generated by the OCR on that site:
'p========r^ SECTION SEVEN CL By W. L. George
THERE is a good old rule which bids us never prophesy unless we know, but, all the same, when one cannot prophesy one may guess, especially if one is sure of being out of the way when the reckoning comes. Therefore it is without anxiety, that I suggest a picture of this world a hundred years hence, and venture as my first guess thrt the world at that time would be remarkable to one of our ghosts, not so much because it was so different as because it was so similar. In the main the changes which we may expect must be brought about by science. It is easier to bring about a revolutionary scientific discovery such as that of the X-ray than to alter in the least degree the quality of emotion that arises between a man and a maid. There will probably be many new rays in 2022, but the people whom they illumine will be much the same. From which the reader may conclude that I do not expect anything startling in the way of scientific discoverv. That is
not the case; I am convinced that in 2022 the advancement of science will be amazing, but it will be nothing like so amazing as is the present day in relation to a hundred years ago. A sight of the world today would surprise President Jefferson much more, I suspect, than the world of 2022 would surprise the little girl who sells candies at Grand Central Station. For Jefferson knew nothing of railroads, telegraphs, telephones, automobiles, aeroplanes, gramophones, movies, radium, &c.; he did not even know hot and cold bathrooms. The-little girl at Grand Central is a blase child; to her these things are commonplace; the year 2022 would have to produce something very startling to interest her ghost. The sad thing about discovery is that it works toward its own extinction, and that the more- we discover the less there is left. % w It does not follow that, scientifically, the year 2022 should fail to be amazing. I suspect that commercial flying will have become entirely commonplace. The passenger steamer will survive on the coasts.
but it will have disappeared on the main routes, and will have been replaced by flying convoys, which should cover the distance between London and New York in about twelve hours. As I am anxious that the reader should not look upon me as a visionary, I would point out that in an airplane collision which happened recently a British passenger plane was traveling at 180 miles an hour, which speed would have brought it across the Atlantic in eighteen hours. It is therefore quite conceivable that America may become separated from Europe by only eight hours. The problem is mainly one of artificial heating and ventilation to enable the aeronauts to survive. The same cause will affect the railroads, which at that time will probably have ceased to carry passengers except for suburban traffic. Railroads may continue to handle freight, but it may be that even this will be taken from them by road traffic, because the automobile does not have to carry the enormous overhead charges
of tracks. Certainly food, mails and all light goods will be taken over from the railroads by road trucks. As for the horse, it will probably no longer be bred In white countries. The people of the year 2022 will probably never see a wire outlined against the sky: it Is practically certain that wireless telegraphy and wireless telephones will have crushed the cable system long before the century is done. Possibly, too, power may travel through the air when means are found to prevent enormous voltages being suddenly discharged in the wrong place. Coal will not be exhausted, but our reserves will be seriously depleted, and so will those of oil. One of the world dangers a century hence will be a shortage of fuel, but It is likely that by that time a
ureal ueai 01 power win ue ooiHineo irom tides, from the sun, probably from radium and other forms of radial energy, while It may also be that atomic energy will be harnessed. If It is true that matter Is kept together by forces known as electrons. It Is possible that we shall know how to dls< ?perse matter so as to release the electron as a force. This force would last as long as matter, thefofore as long as the earth itself. I The movies will be more attractive, as long before 2022 they will have been re* i
THE I P ma
V ' placed by the kinephone, which now exists only in the laboratory. That is the figures on the screen will not only move, but they will have their natural colors and spaak with ordinary voices. Thus, the stage as we know it to-day may entirely disappear, which does not mean the doom of art, since the movie actress of 2022 will not only not need to know how to smile but also how to talk. Hna mttrVif AvianH i ? A ? fi ? i i ?1 vr nn tKa
V/UO "HftUt V<AtVUU 1UUC nunc 1J uu tuo number of inventions which ought to exist and will exist, but the reader can think of them for himself, and it is more interesting to ask ourselves what will be the appearance of our cities a hundred years hence. To my mind they will offer a mixed outlook, because mankind never tears anything down completely to build up something else; it erects the new while retaining the old; thus, many buildings now standing will be preserved. It is conceivable that the Capitol at Washington, many of the universities and churches will be standing a hundred yearB hence, and that they will, almost unaltered, be preserved by tradition. Also, many private dwellings will survive and will be inhabited by Individual families. I think that they will have passed through the cooperative stage, which may be expected fifty or sixty years
nence, wnen ine servant pruoiem nas oecome completely unmanageable and when private dwellings organize themselves to engage staffs to cook, clean, and mend for the groups. That cooperative stage will be the last kick of the private mistress who wants to retain in her household some sort of slave. In 2022 she will have been bent by circumstances, but sh'e will have recovered her private dwelling, being served for seven hours a day by an orderly. The woman who becomes an orderly will be as well paid as If she were a stenographer, will wear her own clothes, be called "Miss," belong to her trade union and work under union rulea. Naturally the work of the household, which is being reduced day by day, will in
2022 be a great deal lighter. I believe that most of the cleaning required to-day in a house will have been done away with. In the first place, through the disappearance of coal in all places where electricity is not made there will be no more smoke, perhaps not even that of tobacco. In the second place I have a vision of walls, furniture and hangings made of more or less compressed papier mache, bound with brass or taping along the edges. Thus, instead of \
VEW YO GAZINE NEW YORK, SUNDAY,
Witt^L
bB
scrubbing Its floors, t^e year 2022 will unscrew the brass edges or unstitch the tapes and peel off the dirty surface of the floor or curtains. Then I every year a new floor board will be J laid. One may hope that standard B chairs, tables, carpets, will be peeled in the same way. SB Similar reforms apply to cooking, a ^B great deal of which will survive ^B among old fashioned people, but a SB great deal more of which will prob- ^B ably be avoided by the use of syn- ^fl thetic foods. It is conceivable, though not certain, that in 2022 a H complete meal may be taken in the m shape of four pills. This is not en- ^ tirely visionary; I am convinced that corned ^eef hash and pumpkin pie will still exist, but the pill lunch will ?roll by their side. But at that time few private dwellings will be built: in their stenH will
rise the community dwellings, where the majority of mankind will be living. They will probably be located in garden Bpaces and rise to forty or fifty floors, housing easily four or five thousand families. This is not exaggerated, since in one New York hotel to-day three thousand people sleep i every night. It would mean also that each ( block would have a local authority of its < own. I imagine these dwellings as afford- 1 ing one room to each adult of the family | and one room for common use. Such cook- l ing as then exists will be conducted by the < local authority of the block, which will also i undertake laundry, mending, cleaning and will provide a complete nursery for the i children of the tenants. i Perhaps at that time we shall have at- j tained a dream which I often nurse, name- 1 ly, the city roofed with glass. That city < would be a complete unit, with accommoda- | Hons for houses, offices, factories and open j spaces, all this carefully allocated. The | root would completely do away with < weather and would maintain an even tern- ; perature to be fixed by the taste of the ( period. Artificial ventilation would sup- ] press wind. As for the open spaces, if the temperature were warm they would ex- l
hibit a continual show of flowers, which < would be emancipated from wlifter and i summer; In other words, winter would not t come however long the descendants of Mr. l Hutchinson might wait. t The family would still exist, even though ] it is not doing very well to-day. It Is in- , conceivable that some sort of feeling be* r
RK HE] SECTl MAY 7, 1922. nT/*A T<^ jo 1
,cF ' w^f^i JBPpl|p^^jMi W. L. GEORGE, the distinguished British autt tween parents and children should not persist, though I am of course unable to tell what that feeling will be. I Imagine that the link will be thinner than it Is to-day, because the child is likely to be taken over by the State, not only schooled but fed and ?lad, and at the end of Its training placed In a post suitable to Its abilities. This may be affected by birth control, which In 2022 will be legal all over the world. There will be stages: the first results of birth control will be to reduce the birth rate; then the State will step in. as it loes in France, and make it worth peo
pie's while to have more children; then the State will discover that it has made things too easy and that people are having children recklessly; finally some sort of balince will establish Itself between the State lemand for children and the national supply. Largely the condition of the family will le governed by the position of woman, be
-ause woman is the family, while man is nerely Its supporter. It is practically cer ;ain that In 2022 nearly all women will lave discarded the idea that they are prlnarily "makers of men." Most fit women *111 then he following an individual career. Ml positions will he open to them and a ;reat many women will have risen high, rhe year 2022 will probably see a large
RALD I [ n
J11Z 38 th 8U
y number of women in Con^ - a press, a great many on the judicial bench, many m . in civil service posts and is perhaps some in the mi President's Cabinet. But it is unlikely that ^ women will have an ^ achieved equality with T1 i.|" men. Cautious feminists At EsS Bk such as myself realize ^f | that things go slowly and ^ that a brief hundred tf,i
r years win noi wipe out im the effects on women of m< 30,000 years of slavery. rr1 ar Women will work, partly m( heeausc they want to and partly because they will so be able to. Thus women w' tin will pay their share in the upkeep of home and t)o family. The above sug- bu gestion of community no buildings, where all the household work will be u' ca; done by professionals, will liberate the average trj wife and enable her out of her wages to pay her ^ share of the household work which she dis- nr1 likes. an Marriage will still exist much as it is Ati to-day, for mankind has an Inveterate taste faf for the institution, but divorce will prob- nn ag ably be as easy everywhere as it Is in Nevada. In view, however, of the lm- Hti proved position of woman and her earning Th power, she will not only cease to be entitled to alimony, but she will be expected, ^ after the divorce, to pay her share of the thl maintenance of her children. or As regards the politics of 2022, I should lsr expect the form of the State to be much / th< the same. A few rearrangements may a8 have taken pla^e on the lines of self-de- or termination; for ?Instance, Austria may Bu have united with Germany, the South pu American republics may have federated, pil Ac., but I do not believe that there will be 1 i a superstate. There will still be republics an and monarchies; possibly, In 2022, the It Spanish, Italian, Dutch and Norwegian tin kings may have fallen, but for a variety of tal reasons, either lack of advancement or oci
inni titu riiim nii'iup, we may export silll vir to And kings in Sweden, .lugo-Slavia, gw Greece, Rumania and Great Britain. on On the inside, these States may have kli slightly changed, for th9ro prevails a ten- an dency to socialization which has nothing to do with socialism. Most of the Euro- tht pean governments are unconsciously na- dif tionalizing a number of industries, and a l
TWELVE PAGES
t i ti wJ Is will go on. One may therefore preme that in 2022 most States will have tionalized railways, telegraphs, teleones, canals, docks, water supply, gas \ ' any) and electricity. Other industries 11 exist much as they do to-day, but it likely that the State will be inclined to ntrol them, to limit their profits, and to bitrate between them and the workers, e find a hint of this in America in the ti-trust acts; a hundred years hence e tendency will be much stronger. It is >rth noting as an international factor
at b/that time purely national Industries 11 almost have disappeared, and that the >rk of the world will be in the hands of ntrolled combines governing the supply a commodity from China to Peru. Unfortunately these international relains through trade are not likely to have ected political conditions. There will ill be war. The wars of that period may a little less frequent than they are toy, and be limited by arrangements such the Pacific agreement, the agreement tween Canada and the United States of rierica to leave their frontier unfortified, ., but it will still be there. I suspect at those wars to come will be made horle beyond my conception by new poison ses, inextinguishable flames and light>of smoke clouds. In those wars the airane bomb will seem as out of date as is
clay tne hatchet. War may ultimately sappear, but this lies beyond the limits this article and even beyond those of f mind. \s regards the United States in particu, it is likely that the country will have me to a complete settlement, with a popition of about 240,000,000. The idea of rth and South, East and West, will have noet disappeared; by that time the Amerin race will have taken so definite a form it immigration will not affect it. The nfirican from Key West and the Amerin from Seattle will be much the same nd of mic. That is to say as regards race, but I feel at mentally the American of 2022 will ve enormously changed. He is to-day e most enterprising creature in the orld, and is driven by a continual urge to se, to make money. That is because the odern American lives in a country that only partly developed, and where imense wealth still lies ready for him to ke. In 2022 that will be as finished as
s to-day in England. American wealth 11 then he eithe.- developed or known, d all of it will belong to somebody, lere will he no more opportunity in nerica than there is in England to-day. lose Americans will know that it is actically certain that they will die much the same position as the one in which ey were born. Those Americans will erefore be less enterprising and much ire pleasure loving. They will have belled against long hours: the chances e that in 2022 few people will work >re than seven hours a day, if as much. The effect of this, which I am sure unds regrettable to many of my readers 11. in my opinion, be good. It was essen1 that the American race should be cable of intense labor and intense ambin if it was to develop its vast country, t one result has been haste, overwork, ise, all of which is bad for the nerves. 2022 America will have made her forae and will he enjoying it as well as she n. I think that she will be a happier counr than she is to-day. The appeal of alth will be less because wealth II be difficult to attain, so those aericans to come will be producing in t and literature infinitely more than they e producing to-day. To-day, in fiction, ? aerlca leads the world by sincerity, tli and fearlessness, hut the American vel of significance is a novel of revolt
alnst the thralls of money, of convenn anrl of puritanism. In 2022 Ameriran srature will be a literature of culture, e battle will he over and the muzzle There will be no more things one n't say, and things one can't think No ubt there will be In 2022 people who nk as they would have thought in 1022, even a little earlier, but a great liberaln of mind will prevail, rt is not my business to corgrat ilat" > future, and I have no desire to do so. It Is impossible to say a thing Is good bad; all one can say is that it exists, it in case some of my readers feel relsion when they contemplate my lunch Is or my nationalized railroads, to those vould say that they are perhaps unduly xious. The world takes care of Itself; has been doing so for hundreds of cen
ries nnri is still spinning; tne worm win te care of Itsplf In 2022: that Is Its chlpf r-upatlon. Morp than that. I fppl conlcpd that though thp world may Iosp icps, It will dpvolop othpr grarps, that the wholp. and as timp goes on. man id grows more intelligent, more amiable d more honpst. rhp fnturp will bp difficult: what dops it mattpr? So was thp past difficult: flcultlps did not prevent Its turning into tolerable preser^.
I love the idea of having everything in one language; dealing with just one thing instead of HTML, CSS, and JavaScript.
But isn't that why everyone moved over to React, Angular, and Vue? It's nice that this is in WASM but honestly, I think I prefer React components written in TypeScript, and you can probably get that compiling to WASM with AssemblyScript.
My two cents: this is a super neat hobby project! Probably won't ever make it to production anywhere though.
No, JSX in AssemblyScript with the same dev experience is very far away.
AssemblyScript is awesome, but right now it's more similar to C (with TS syntax) - it's not a TypeScript runtime on Wasm, it's a lowlevel language that looks nearly the same, that hopes to integrate Wasm GC etc when it comes (not yet).
Static TypeScript is much closer to regular TypeScript. But it doesn’t support JSX either (and I while it might make sense with a custom pragma it certainly wouldn’t make sense with React as a default).
That's ultimately why I couldn't move forward in it. My brain just kept thinking typescript and it was hard to manage the cognitive dissonance of it being identical syntax but with invisible limitations I wasn't used to.
Please let's not kid ourselves. Propaganda will work on most populations. Somewhat better in places where literacy is low, but it can absolutely take hold anywhere.
There are useful idiots in every population ready to tattoo themselves with some thing either political, ideological, or even commercial.
Populations are not good at context - we take some tiny, possibly insignificant effect, like the utterance of a leader etc. and make all sorts of giant assumptions about it. Sometimes the issue is serious but we still contextualize it.
We also tend to ignore major material issues unless someone is creating a fuss and making a narrative about it.
> P.E.I. is suspending its use of the AstraZeneca-Oxford COVID-19 vaccine for 18- to 29-year-olds.
Not only that, but:
> Quebec, Ontario, Alberta, Saskatchewan, Manitoba, Newfoundland and Labrador and Prince Edward Island have all suspended the use of the vaccine for anyone below the age of 55. Other provinces and territories are expected to follow.
Please correct me if I'm mistaken, but I did check a couple of years ago and found what I thought were some errors. I began to draft an email to you but never finished and eventually discarded it. I'll try to recreate it here:
With regards to halting, there are some programs that can be written simply in any common programming language to halt iff a solution to a specified number exists, simply by enumerating all possible values and halting if one satisfies the neccessary conditions. Thus to determine the halting of such a program would solve a problem. Typically these can be done in less than ten lines of Python, for instance. Some examples include: does there exist an odd perfect number; do there exist four positive integers a,b,c,n with n>2 such that a^n+b^n=c^n? The former is an open question; and the latter is Fermat's Last Theorem, which took an immense amount of machinery to prove, six decades after Turing's paper, and which is related to many open problems. (I've also heard that the halting problem could be used to solve the Riemann Hypothesis, but I don't know whether or not that's true.) If you can find a way to figure out whether or not such simple programs halt or not, I'd love to hear about it! What I've gleaned from your papers doesn't actually seem to apply to such programs though, which is a shame since these programs aren't very complicated.
Now, with regards to Cantor, I recall reading a paper on your website (which I can't seem to find today, so apologies if I misrepresent it). I agree that set theory is not a good way to evaluate the number of numbers, since it throws out all properties about such numbers (e.g. ordering); this is why we have natural density, real density, and other properties. But there was another part where you indicated (something to the effect of) that the reals were countable since there are countably many Turing Machines; however I believe the correct conclusion to draw is that there are uncomputable numbers. Again, I can't seem to find it today, so I apologize if this is unsubstantiated.
You point out quite correctly that if halting can be solved, then we can use it to settle certain open mathematical questions. But this does not tell us whether halting can be solved. (And if it can, it may not be a practical way to settle those questions.) I have many papers on the halting problem on my website at www.cs.utoronto.ca/~hehner/halting.html. The most recent is www.cs.utoronto.ca/~hehner/EGT+.pdf published in SN Computer Science v.1 p.308, 2020 September. Its final paragraph says:
Let X and Y be two programming languages, or two computers, or two locations. It is inconsistent to ask for an X-program to compute halting for all X-programs due to a twisted self-reference. Twisted self-reference is characteristic of subjective specifications. So it may be consistent and satisfiable to ask for a Y-program to compute halting for all X-programs. At least it has not yet been proven impossible.
If you want to know what a twisted self-reference is, or what a subjective specification is, you have to read the paper. This conclusion is not what you claimed it is. The Cantor paper you are unable to find is on my website. Its name is "the Size of a Set", and it is at www.cs.utoronto.ca/~hehner/SetSize.pdf. Its conclusion says:
It is popularly believed that Cantor's diagonal argument proves that there are more reals than integers. In fact, it proves only that there is no onto function from the integers to the reals; by itself it says nothing about the sizes of sets. Set size measurement and comparison, like all mathematics, should be chosen to fit the needs of an application domain. For all application domains that I know of, Cantor's countability relation is not the most useful way to compare set sizes.
I have never claimed, as you said, that there's anything wrong with Cantor's diagonal argument. My paper on Goedel, published in Beauty is our Business, Springer-Verlag silver series, New York, p.163-172, 1990, is at www.cs.utoronto.ca/~hehner/God.pdf. Nowhere in that paper is there any suggestion that "Goedel was wrong", as you claimed I said.
> So it may be consistent and satisfiable to ask for a Y-program to compute halting for all X-programs. At least it has not yet been proven impossible.
Alas, it has been proven. RAM computers are equivalent to Turing machines. If X and Y are Turing-complete languages then all X-programs can be converted to Turing machines, your proposed Y-program can also be converted into a Turing machine. Put another way, unless your language X is too weak to be useful, there should exist a deterministic program to convert any program written in Y into a program written in X with identical behavior (with regard to halting).
This should be evident by the fact that the computers we use day-in and day-out execute one binary machine language. Any X-program and Y-program reduce to the same assembly language.
The twisted self-reference must therefore be language-agnostic. You can absolutely solve halting for various simplified "languages", but not Turing-complete languages.
Here's a sketch of a proof explaining why this is impossible (under the Church-Turing thesis).
Suppose you have written a Y-program y that takes an X-program x as input, and computes whether or not x halts (say, on no input), AND y halts every time.
If X is a Turing-complete language, then for a given Turing Machine t we write an X-program xt that simulates t and halts iff t does. Then we run y on xt. This decides whether or not t halts, and works for all TMs t; therefore y solves the halting problem for all Turing Machines. But we know from Turing's "twisted self-reference" that no Turing Machine can do this, so the language Y must be strictly more computationally powerful than the Turing Machine model. This would disprove the Church-Turing thesis, which would be pretty incredible if true, but seems highly unlikely (since modern computers, and even quantum computers, are polynomial time equivalent to TMs).
Otherwise, X is not a Turing-complete language. This is valid, I guess, but not really relevant to the problem of halting, since such a language would be very underpowered for most useful purposes. Now, regular languages are decidable, which is great for lots of computations that can be encoded using predicate logic and first-order arithmetic. Even climbing one rung up the computational complexity to context-free languages brings undecidability, though, since many problems (e.g. whether or not two context-free grammars generate the same language) are undecidable.
Regarding Goedel, I apologize and retract my claim.
Regarding Cantor, indeed, there are many ways to compare the "sizes" of two sets. Cardinality is one such way. It can have applications, for instance Cantor's original paper used it to prove that there are infinitely many transcendental numbers. It is also used in real analysis, in Lebesgue integration, and in measure theory. But it is not the only way to do so. There is a very real sense in which {1,2,3} is a smaller set than {4,5}; that does depend on knowledge of the contents of the set though, not merely the number of elements contained. In a more abstract context where measure or natural density aren't applicable, cardinality can be left as the only well-defined definition of "size", but in most contexts I agree, it's just one of many possible definitions and they should each be used with care.
Arguing about the sizes of different infinite sets is tricky, though. The set of even integers has the same cardinality as the set of all integers, even though it has density 0.5. The cardinality of the set of rationals is somehow "infinitely less" than the set of irrationals, even though both sets are dense on one another. But there are some things we can learn from not having
Me, however, I like cardinality for the aesthetic properties. I like the Axiom of Choice and all the nonsense that follows from it. Cardinal infinities behave quite differently from the well-defined infinities of ordinals, hyperreal numbers, and surreal numbers.
In your article "The Size of a Set", you discuss Formalism. Are you aware of the axiom "V=L"? It's quite neat, because by effectively outlawing any non-constructible sets, the Generalized Continuum Hypothesis follows! If you accept cardinality as a purely aesthetic definition in conjunction with Formalist ideology, you get a very beautiful result. Practical, maybe not, but beautiful.
That is all to say that I agree with a lot of what's in your paper on "the Size of a Set". However, the "Program Analogy" section of the paper contains what I think is a flawed argument:
> [A] Cantor-type conclusion concerning the sizes of sets of programs is absurd. The proper conclusion is simply that there is no program (in some programming language) to generate all and only the infinite-sequence programs (in that same programming language). It is not a conclusion about the sizes of sets. Likewise in the original Cantor argument, the proper conclusion is not about the sizes of sets; it is simply that there is no sequence of all infinite sequences.
I think the correct conclusion here is that to determine which programs are "infinite-sequence" is undecidable. You've reached a contradiction, therefore the last assumption must be flawed; in this case the assumption was that you could write a program D "whose execution neither halts nor goes into a non-printing infinite loop." This assumption is about programs and programming languages, and says nothing about sets, so the analogy is flawed. In Cantor's argument, the assumption is that the set [0,1) can be ordered. You're correct in saying that this is merely a statement about the existence of a bijection between the naturals and the reals, but that is the definition of cardinality. Your issue here is that you don't like saying that cardinality means "size", and that's fair. But I think this analogy is a stretch at best, and irrelevant to your overall argument.
I said: So it may be consistent and satisfiable to ask for a Y-program to compute halting for all X-programs. At least it has not yet been proven impossible.
You said: Alas, it has been proven. RAM computers are equivalent to Turing machines. If X and Y are Turing-complete languages then all X-programs can be converted to Turing machines, your proposed Y-program can also be converted into a Turing machine. Put another way, unless your language X is too weak to be useful, there should exist a deterministic program to convert any program written in Y into a program written in X with identical behavior (with regard to halting). This should be evident by the fact that the computers we use day-in and day-out execute one binary machine language. Any X-program and Y-program reduce to the same assembly language.
I say: You cannot have read my papers carefully. I spent a lot of time explaining, with examples, why programs with subjective specifications cannot be translated from one TM-equivalent language to another TM-equivalent language.
You said: This would disprove the Church-Turing thesis.
I say: Again, you cannot have read the paper carefully. There is a section in each of www.cs.utoronto.ca/~hehner/OSS.pdf and www.cs.utoronto.ca/~hehner/EGT+.pdf titled "Church-Turing Thesis", explaining that it applies to all objective specifications, which are the only specifications considered by Church and Turing, but not to subjective specifications.
You said: Regarding Cantor, indeed, there are many ways to compare the "sizes" of two sets. Cardinality is one such way. It can have applications, for instance Cantor's original paper used it to prove that there are infinitely many transcendental numbers. It is also used in real analysis, in Lebesgue integration, and in measure theory.
I say: Again, you cannot have read the paper carefully. Almost every time I mention applications in the paper, I say "applications outside mathematics". If you consider mathematics to be an application of mathematics, then all mathematics has applications.
You said: the "Program Analogy" section of the paper contains what I think is a flawed argument.
I say: Yes, I point out the flaw in the paper in the paragraph that starts with the sentence "Like the original Cantor argument, this program-analogy version is informal, and the informality may hide serious errors.". Did you miss it?
You will "win" this debate by throwing misunderstandings at me faster than I can reply. And it's disheartening because the replies are already in the papers, if you read them carefully.
> I say: You cannot have read my papers carefully. I spent a lot of time explaining, with examples, why programs with subjective specifications cannot be translated from one TM-equivalent language to another TM-equivalent language.
> I say: Again, you cannot have read the paper carefully. There is a section in each of www.cs.utoronto.ca/~hehner/OSS.pdf and www.cs.utoronto.ca/~hehner/EGT+.pdf titled "Church-Turing Thesis", explaining that it applies to all objective specifications, which are the only specifications considered by Church and Turing, but not to subjective specifications.
Fair point, I had not read OSS. That said, I think your interest in "subjective specifications" goes against your apparent disinterest in the "twisted self-reference" of Halting Problem proofs. It throws readers off as you attempt to add twisted self-reference to the domain of programming, where it isn't usually found, and simultaneously remove it from other places where it emerges naturally (as a consequence of the Church-Turing Thesis).
> I say: Again, you cannot have read the paper carefully. Almost every time I mention applications in the paper, I say "applications outside mathematics". If you consider mathematics to be an application of mathematics, then all mathematics has applications.
Fair point, I concede.
> I say: Yes, I point out the flaw in the paper in the paragraph that starts with the sentence "Like the original Cantor argument, this program-analogy version is informal, and the informality may hide serious errors.". Did you miss it?
No, I didn't miss it, but the inclusion of such an irrelevant and flawed analogy is perplexing and makes the paper harder to read, and seem less worth the time it takes to read carefully. You've written a lot, and some of the material can be tough to chew through (especially that written with the structure of formal logic).
> And it's disheartening because the replies are already in the papers, if you read them carefully.
I've reread your OSS and EGT papers, with the additional help of your comments above, I think I now understand your stance better. I think part of your notoriety comes from your papers appearing to contradict the Halting Problem, with sections titled "How to Solve Halting" that don't actually provide a clear method. Instead, emphasizing that you're examining self-referential logic puzzles reminiscent of those found in Raymond Smullyan books would help clarify what you're doing. Many readers come in with the Bernard Russell mindset of "Let's try to avoid these self-referential questions" and thus have trouble adjusting to your angle. Most readers interested in The Halting Problem are not interested in "subjective specifications."
Your papers aren't easy reads (and I'm not complaining: it's an unavoidable consequence of their nature as papers on self-referential logic) and a brief first pass of your papers leaves many of us misinterpreting your angle. I think some editing of your introductions and conclusions could help immensely. Be very upfront and explicit that you don't disagree with the common conclusions of The Halting Problem for objective specifications, but instead that you're more interested in exploring a different side of programming. (In this regard, Jeffrey Shallit's mean-spirited "fringe computer science" comments make sense, but really ought not to be an insult since you are in fact exploring an oft-ignored side of the field.) Err on the side of caution, and over-communicate your intent with the papers! When many readers think you've said X when you have not, in fact, said X, take that as constructive feedback to more explicitly say "not X" in edits and future papers.
Thank you for guiding me in reading your papers. Your arguments in this thread have helped me make sense of them. I think I finally understand your arguments, and I hope with a change to your approach, you can successfully reach other readers too.
Dunno how much I trust this guy. He's notorious for claiming that the Halting Problem is actually solvable, and in other places suggests that Godel was wrong, and even that Cantor's proof that the reals are uncountable is wrong. http://www.cs.toronto.edu/~hehner/halting.html Finding the flaws in all those arguments is a good challenge if you have a free afternoon!
