S-Convexity Mysteries of Faith

When the initial premises of mathematical proofs are wrong, we cannot get a mathematical proof because the logical system we rely on, when doing Mathematics, is not Classical Logic, but a subset of Classical Logic, which is Mathematical Logic instead. In this way,  a value false for the antecedent cannot bring a true implication, and the possible proof originating in it cannot be told to be a proof. 

It is wrong because if you take alpha to be 0.1, beta to be 0.5, A to be 0.2, and B to be 0.3, the inequality does not follow from the premises. 

In this case, most people would not keep on reading the paper, since that is almost the first line on it that matters in the communication of the results.

Combinatory Logic: A Quick Take (being built)

Self-membership, and self-application are terms associated with the reasons for the creation of the branch of logic designated by the expression Combinatory Logic (Bimbo 2016, para. 6).

Self-membership is the central topic of the Russell’s Paradox, and this paradox is considered unsolvable by many experts (Sion 2017, p. 130).

Combinatory Logic has been invented by Shönfinkel, and it was developed by Curry in the 1920s (Baker 2019, para. 2). Both of them are classified as mathematician-logician [(Fracademic 2019), (Seldin 2019)].

             Moses Schönfinkel                    Haskell Brooks Curry

                    (1889-1942)                                  (1900-1982)

          Born in Dnipro, Ukraine           Born in Massachusetts, US

  

[(Pngkey.com 2018), (Fracademic 2019)]         (Seldin 2019)    

In Combinatory Logic, any expression can be combined with any other expression (Baker 2019, para. 2).

Its special symbols are ‘B’, ‘I’, ‘K’, ‘S’, and ‘W’ [(Seldin 2019, para. 34), 

(Baker 2019, para. 2)], and these letters seem to originate in the German language: 

Identität/Identity (I), Wiederholen/Repeat (W), Konstanten/Constant (K),  

Substitutionsprozesse/Substitution processes (S), and Beherbergen/Take in (B) 

[(Curry, H 1930), (Google.com 2019)].

It also uses parenthesis, and variables, and its variables are single characters, such as ‘x’, and ‘y’ (Baker 2019, para. 2).

I is the identity operator, and it is defined to be \x x [(Seldin 2019, para. 4), (Pryor 2015, para. 3)].

K is the constancy operator, and it is defined to be \x y. x: K eliminates its second a

rgument [(Seldin 2019), (Pryor 2015, para. 4)].

S is the distributor, and it is defined to be \f g x. f x (g x): S copies its third argument, and distributes it over the arguments that precede it [(Seldin 2019), (Pryor 2015, para. 5)].

B is the composition, and it is defined to be \f g x. f (g x): B changes f into a function, and then g, and x into its arguments (Pryor 2015, para. 8).

W is defined to be \f x. f x x: W doubles x (Pryor 2015, para. 11).

References

Bimbo, K 2016, Combinatory Logic, Stanford, viewed 1 August 2019, <https://plato.stanford.edu/entries/logic-combinatory/#ReduEquaTheiForm>

Barker, C 2019, Combinatory Logic Tutorial, viewed 1 August 2019, <http://www.nyu.edu/projects/barker/Lambda/ski.html>

Pryor, J 2015, Week3 Combinatory Logic, Combinators and Combinatory Logic, viewed 1 August 2019, <http://lambda.jimpryor.net/topics/week3_combinatory_logic/>

Pngkey.com 2018, File – Schonfinkel – Moses Schonfinkel, viewed 1 August 2019, < https://www.pngkey.com/detail/u2e6a9t4w7y3q8u2_file-schonfinkel-moses-schnfinkel/>

Seldin, J 2019, Haskell Brooks Curry, viewed 1 August 2019, <https://www.iep.utm.edu/curry/>

Fracademic 2019, Moses Schonfinkel, viewed 1 August 2019, <https://fracademic.com/dic.nsf/frwiki/1190889>

Sion,  A 2017, Paradoxes and Their Resolutions: A Thematic Compilation, self-published, Geneva, viewed 1 August 2019, <https://books.google.com.au/books?id=nqg-DwAAQBAJ&pg=PA130&lpg=PA130&dq=paradox+self-membership&source=bl&ots=6ea0W6hs7v&sig=ACfU3U1IqRx8Hf5AkwDLXbVexvTOvM9gNQ&hl=en&sa=X&ved=2ahUKEwibz9bK6eDjAhWUWX0KHbZJBa8Q6AEwC3oECAgQAQ#v=onepage&q=paradox%20self-membership&f=false>

Curry, H 1930, “Grundlagen der Kombinatorischen Logik”, American Journal of Mathematics, vol. 52, no. 3, pp. 509-538, viewed 8 August 2019, <https://www.jstor.org/stable/2370619?read-now=1&seq=5#page_scan_tab_contents>

Google.com 2019, Google Translate, consulted 8 August 2019, <https://www.google.com/search?q=google+translate&rlz=1C1GCEU_enAU820AU820&oq=google+translate&aqs=chrome.0.69i59j69i64l2j69i60.4245j0j8&sourceid=chrome&ie=UTF-8>

Factoring Challenge

Factoring

Factoring is an important part of Cryptography and Security, so that it makes perfect sense finding this topic here. 

The unique factoring of a number makes concealment easy: if a person knows the key, so say the factors involved, they should be able to de-code the message in a relatively easy way.

We want to pass a message from one end to another during a war, and the enemy can only de-code it in reasonable time if they have our keys. 

They used to give money to individuals in exchange for the factoring of a number. 

From the mentioned source:

"Starting in 1991, RSA Data Security offered a set of “challenges” intended to measure the difficulty of integer factoring. The challenges consisted of a list of 41 RSA Numbers, each the product of two primes of approximately equal length, and another, larger list of Partition Numbers generated according to a recurrence.

The first five of the RSA Numbers, ranging from 100 to 140 decimal digits (330–463 bits), were factored successfully by 1999 (see [2] for details on the largest of these). An additional 512-bit (155-digit) challenge number was later added in view of the popularity of that key size in practice; it was also factored in 1999 [1].

In addition to the formal challenge numbers, an old challenge number first published in August 1977, renamed ‘RSA-129’, was factored in 1994 [1].

The Quadratic Sieve was employed for the numbers up to RSA-129, and the Number Field Sieve for the rest."

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