Friday, 15 July 2022

A Response to Mark Dodds

Sometimes we have the right intuitions, and we even publish those, as Mark Dodds has done on https://www.cantorsparadise.com/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793. He then used expressions such as 'crazy claims' to refer to some of the findings of Ramanujan and Grandi.

 

As the author of Infinis (http://www.protosociology.de/Download/Pinheiro-Infinis.pdf), I would like to say that infinity is not a place: it is the absence of one. Things being such, we cannot call it a name, and use this name to make calculations. Mathematicians know: we indicate infinity, but we don't use it in mathematical operations. 

 

If you have a false premise, there is a good chance you will reach a false conclusion. It is not guaranteed to be the case, since, for instance, if you assume that 0 is equal to 1/1 and you then swap / for -, you get 0=1/1=1-1=0, or 0=0, therefore a true conclusion. 

 

An implication that has a false antecedent will always be true in Mathematics, as we see on http://article.sapub.org/10.5923.j.ap.20170101.03.html, so that, in mathematical proofs (taking away the proof by contradiction), we always have a true antecedent instead of a false one. A true antecedent with a false consequent gives us a false implication. A true antecedent with a true consequent gives us a true implication. Now it makes sense, since we can tell wrong from right… .  

 

Now, the partial sums in (1-1)+(1-1)+(1-1)...diverge, since S1=1, S2=0, S3=1, S4=0, and so on. The partial sums in 1+2+3+4... also diverge, since S1=1, S2=3, S3=6, S4=10, and so on, so that we know there is no 'place of arrival' for these sums, and therefore we can never equate them to a certain place, so say A, B or C. There should be no equal sign there! If we put an equal sign after the sums (1-1)+(1-1)+(1-1)+…, and 1+2+3+4+…, that is a false assumption, a false antecedent. We can only have true ones (it is not a proof by contradiction). 

 

Since we assumed those sums had a 'place of arrival', we got to true or false conclusions, and to true implications, what made us have the sensation that we had a proof, but it was actually just a hoax. 

 

If we deal with the sums on their own, without adding an equal sign to them, and a 'place of arrival' or variable, we get to the right conclusions, which are that they both go to nowhere in particular. 

 

Notice that, in reality, in Mathematics, we can put an equal sign there, but we then call the result sum or limit of an expression when n goes to infinity, what will make the conclusions of Ramanujan and Grandi impossible because we then have ∑(-1)n=1-1+1-1+… therefore 1-∑(-1)n=1-(1-1+1-1+…) therefore 1-∑(-1)n=1-1+1-1+… therefore 1-∑(-1)n=∑(-1)n therefore ∑(-1)n+∑(-1)n=1, but ∑(-1)n diverges, and therefore we cannot find a result for  ∑(-1)n+∑(-1)n, as we see on https://math.stackexchange.com/questions/59512/sum-of-infinite-divergent-series, and therefore ∑(-1)n+∑(-1)n is not 2∑(-1)n, and therefore the ‘crazy’ result does not eventuate.


Thursday, 8 August 2019

Combinatory Logic: A Quick Take


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>

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>






Tuesday, 22 January 2019

Factoring Challenge









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."


Thursday, 20 December 2018

Graph Theory: Pure or Applied Mathematics?




Graph Theory worries about the study of graphs. That connects nicely to Topology and Geometry and therefore to what we traditionally call Pure Mathematics. 


For those who have doubts:


Watch:





Sunday, 11 June 2017

Mobius Band






Band brings a woman, a bagel, and, possibly (not my area), a huge inaccuracy: two sides and one surface... 


A band, by definition, has two faces: if we flatten it up, then we have upper and lower or over and under, whatever we want to call them, but it is still two sides. 


I liked the idea of the bagel being cut like that...


It is a short video.


I think it has a huge inaccuracy. 


The knife is not lifted? 


Do we lift the knife when spreading cream cheese on a bagel that is opened traditional way?


Why would we invest so much time to open a bagel if we can simply do it in a normal way?


Only female... 1 minute and 47 seconds it says... record time, shortest ever, or something like that.


Apparently, we say face in Mathematics, not side. 


In this way, we could stick to the normal definition of side. 



This defines side as a surface (American Heritage Dictionary). 


According to the same source, the Collins English Dictionary says that side is face. 


In this case, she might be right sometimes. 


We say that we are measuring surfaces when we calculate area, so that we must think of how to calculate the area of the band. 


In this case, it would be only one surface, but then all other figures would have only one surface, since we flatten them all before calculating area. 


Surprisingly, Wolfram, which used to be my preferred source for Mathematics in 2001, does accept her definition of side, and then says that the band has one side. 


See: Wolfram



"The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323)."





A few inaccuracies inhabit Wolfram's pages... 



Side cannot be the top or the bottom face, therefore any face but top and bottom, so that the band could also have no sides if we go with what people saynot top or bottom




The best argument I have to state that the Mobius Band has two sides or two faces is that we could use the same reasoning, of the walk, for any common solid, so say a prism.


 We start with the finger on one face and we go all over the figure coming back to the same place. 


That cannot mean that all those faces became one. 


It cannot... 


Face is about the aspect: As the dictionary says, it is what we see in front of us. 


In this case, we see two faces or two sides in the untouched band: The inner and the outer. 


We would have to have at least these two sides or faces to form a three dimensional figure, since only two dimensions could show one-sided things, the flat shape in the Cartesian Plane. 


Prism brings a nice collection of prisms. Say we choose this one:



Now, do the same: Run your finger, index finger, over the lateral surface of this prism (and notice that top and bottom is relative: put it standing and what was side according to one of the sources (not top or bottom) will become a non-side). 


And now? Do we have only three sides? 


You could run your finger over four of the surfaces and come back to the same place... 


That is not a good argument... 


I found no definition relating finger walk to sides or faces, so that not even the non-mathematical sources support this view. 


We must have a mathematical agreement on what a side or face is, however. 


We cannot define face based on what we had before connecting elements, before forming the shape, since otherwise all shapes would be the initial flat surface, and therefore would have one face. 


Notwithstanding, any three-dimensional shape, perhaps taking away the line, would have to have at least two faces, for otherwise it would fit the Cartesian Plane and it would be in 2D instead. 


The definition should be visual, and therefore based on the source that said it is what we see in front of us: if we look at the band from the front, we see one face. If we look at the band from the back, we see another, to the back of the face we just mentioned. 


We put a number on what we see and we will have two. 


For instance, take the picture that follows (it came from band):



Write 1 in all you see when starring at this. 


Now put it upside down and write 2. 


Any other angle will return the same numbers, so that we would definitely have 2 in the end.


Perhaps we define it as being the largest number we may have when changing angles of sight of a 3D-shape. 


In this way, our selected prism would be positioned in a way to show one face at a time to us, so that we get six. 


Where we have an edge, we have the encounter of two faces. We can then just count the edges of the band. 


We definitely have two: The top of the band and the bottom.


We also have the same faces meeting there, so two. 


Got the idea from here (edge):

See:



Sunday, 9 April 2017

Circumferences








YouTube brings a SAT question. 


We have a circle of radius that is 1/3 of the radius of another circle. 


They ask how many times the smaller circle goes around the bigger circle. 


The answer should be 3: 2 Pi r/3 would be the length of the circumference of the smaller circle. 


With this, we need to multiply it by 3 to get 2 Pi r, which is the circumference of the bigger circle. 


That means that the length of the smaller circumference will mean 3 turns over the bigger one for it to go back to the initial point. 


Please write to drmarciapinheiro@gmail.com if you want to converse about any of the contents of my blogs here.





Saturday, 8 April 2017

Pizza and Mathematics







Pizza brings an interesting question and an even more interesting correction of the student's answer: the student seems to have used good logic. 

I thought in the same way, to be sincere. 

It is confusing. 

Marty is told to have eaten 4/6 of his pizza. 

Luis is told to have eaten 5/6 of his pizza. 

Marty ate more pizza than Luis. 

How is that possible? 

The student answered: Marty's pizza was bigger. 

That sounds really logical: You just have a larger radius for this pizza, and therefore his 4/6 ends up being more value in pizza than Luis' 5/6. 


If you do not specify to the level you are thinking, the student has to win on this one. 


If the intentions were saying that that was unreasonable, as the presenter states, the teacher would have to have written pizzas of the same size. 


It says it is about being reasonable. 


When you ask us why, reasonable is assuming that whatever you described is a fact, has already happened, not that you are lying or inventing. 


Reasonable has to be where the average thinker goes with their thinking when reading. 


Maybe those who know Mathematics would think like the boy did...