infinis que sera tamen

8/6/2019 Infinis que sera tamen

http://slidepdf.com/reader/full/infinis-que-sera-tamen 1/24

Infinis que seras tamen1

I. M. R. Pinheiro2

Abstract:

A new system of both geometric representation and reference for

mathematical objects is introduced. One of the many possible reasons

for swapping the Cartesian Plane system for the system presented in

this paper is lightly discussed in the own paper: The amount of

scientific mistake that has originated in the perpetuation of certain

erroneous paradigms, which ought to have appeared through

subliminal processes deriving from the use of the Cartesian Plane as a

mathematical tool.

Key-words:

Infinity, Cartesian, coordinates, space, time, world, number, limit,

reals, operations.

1 Copying the statement libertas que seras tamen, translated into freedom, even if

late, and adapting it to our notion: No borders, even if late… .

2 Po Box 12396, A’Beckett st, Melbourne, Victoria, 8006. E-mail:

[email protected].

Infinis que seras tamen 1/24



1. Introduction:

The term infinity apparently comes from the Latin word infinis, which

means no borders (see [S. Schwartzman 1994], for instance).

Neglecting the origins of the term, mathematicians have said that

there was an infinity of numbers between each randomly chosen

couple of rational numbers, for instance.

It is obvious, however, that the infinity the never-ending progress of

the horizontal Cartesian axis forms is of different nature -completely

different- from that infinity of numbers between any randomly chosen

couple of rational numbers: The in-between numbers are obviously

bounded by at least the couple of rational numbers that have just

been mentioned, for instance.

Basically, when we write about the infinity of numbers between two

rational numbers, we a sort of imagine an infinity that fits some very

well limited length, for which starting and finishing points are well

controlled by us.

Truth is that if both abstract objects ever deserved the same pointer,

such could only be in the English language, never in Mathematics.




How could something that is reached inside of something else

compare to something nothing ever reaches?

Infinity is obviously a referent in the human language, but that does

not necessarily imply that infinity may be used as a referent in

Mathematics.

For each and every referent that we decide to insert in the world of

Mathematics, we need to prove to ourselves that the condition points

to a single world reference, with no mistake or confusion is satisfied,

once Mathematics has to be the scientific place of maximum amount

of control (over all inside of it).

The linguistic referent infinity, unfortunately, as we shall endeavour to

prove in this paper, is one of the referents that does not satisfy the

just mentioned condition.

That does not mean that the referent infinity is useless, just means

that it is not an acceptable mathematical referent.

Infinity is still a respectable referent, which has inspired several

people both inside and outside of Mathematics.

Someone in Mathematics, for instance, felt compelled to create Aleph

to represent the last number, which would not be infinity, but would

correspond to it (?)… .

Such a fact is found reported in [H. C. Parr 2003] together with a very

elucidative discussion on that. According to the source, Aleph-0 would

correspond to the order of the infinity to which the natural numbers




would belong: A trial of measuring how much more infinite one infinity

is in comparison with another, matching our intuition, which tells us

that there are more rational numbers than natural numbers in the

world of Mathematics, for instance. The same source, however, brings

trivial argumentation as to why Aleph-0 is not a good measure for

how big the infinity of the natural numbers is: They multiply each

natural number by two and obtain the even numbers, claiming that to

be half of the previous size and still of Aleph-0 size... .

The discussion that we have just described is one more piece of

evidence about there being something absolutely wrong either with

the term infinity being used in Mathematics or with the world

reference chosen to describe infinity in Mathematics.

Aleph makes us realise that we need more than one term of the

human language to designate the current mathematical references

for infinity, once the current lingo leads people to at least sometimes

imagine that there is a cardinality attached to the concept of infinity,

cardinality that is of fixed size, like the current lingo leads people to

sometimes imagine that infinity is a physical length.

We would like to state that the right question to be asked is why?, not

how?, as for starters, in any theory.

The motivation for creating the denomination infinity for the idea of

never finding an end in the horizontal Cartesian axis is obviously

different from the motivation for creating a denomination for the idea




step size that we take over them (one unit, half unit, and etc.). This

way, walking in the positive direction on the real numbers line would

lead the walker to the conclusion that the only possible answer to the

question what is the highest real number? is real numbers bear

(infinis) no borders, that is, they go nowhere, or they keep on going

forever. Because mathematicians are lazy, instead of stating real

numbers have infinis, they say they go to infinity, meaning the

special place where no borders exist, or no place at all, once there is

never a stopping point.

Infinity, according to one of our most popular dictionaries ([Merriam-

Webster-1 2009]), would mean:

Main Entry:

in·fin·i·ty

Pronunciation:

\in-'fi-nə-tē\Function:

noun

Inflected Form(s):

plural in·fin·i·ties

Date:

14th century

1 a: the quality of being infinite b: unlimited extent of time, space, or quantity :

boundlessness2: an indefinitely great number or amount <an Infinity of stars>3 a: the

limit of the value of a function or variable when it tends to become numerically larger

than any preassigned finite number b: a part of a geometric magnitude that lies beyondany part whose distance from a given reference position is finite <do parallel lines ever

meet if they extend to Infinity> c: a transfinite number (as Aleph-null)4: a distance so

great that the rays of light from a point source at that distance may be regarded as

parallel

However, there is substantial difference between a precise place

called infinity (meaning number 3a from the dictionary extract) and a


http://infinite/

http://infinite/

http://boundlessness/

http://boundlessness/

http://infinite/



quality meaning bearing no borders (meaning 1b from the dictionary

extract).

It has to be the hugest mistake of all stating that the real numbers

keep on going forever, go to infinity . No, they keep on going forever ,

never stopping, at most with infinis.

Notice that the set of the real numbers does not have any of the

borders, but the set of the natural numbers, for instance, does not

have the right border only, so that we should have two different

referents, one for each one of the mentioned situations, if willing to

apply human language to Mathematics, for, in Mathematics, no

inaccuracies are allowed: If a term generates confusion (confusion

might simply mean multiple interpretations), it is simply not good

enough to be part of the mathematical lingo.

Given so much inaccuracy in the Mathematics that is generated by

the use of the term infinity, the most sensible attitude of all is erasing

the concept, as well as the symbol, for infinity everywhere in

Mathematics and starting over.

In Mathematics, the symbol ∞ and the word infinity are applied to at

least the following situations:

a) Mathematical description (and calculation) of limits;

b) Mathematical description of intervals; and

c) Primary school mathematical operations.

The limit of the function f(x) = x , when x never stops growing (or




when x goes to infinity , according to the language used so far in the

world of Mathematics) in the set of real numbers, to use one of the

situations that we have just mentioned, has to be delusional, for there

is no accumulation point in the real numbers line that does for the

end of the set as a whole.

Therefore, the right decision here would be excluding, from the realm

of possible questions in Mathematics, what is the limit of the real

function f(x) = x when the domain member grows to infinity?.

Basically, we need to state, in this case, that the limit of f(x) = x, as x

never stops growing in the set of the real numbers, is not a valid

mathematical request because there is no accumulation point at the

end of the real numbers line (writing things as they are is obviously

the most basic mathematical duty of all).

Notice that we had to appeal to the use of common language,

therefore to the use of terms that are not part of the mathematical

lingo, to explain what was going on in the just mentioned situation.

The explanation is simple: We have proved that the just mentioned

request is not sound in Mathematics, therefore does not belong to the

world of Mathematics. If something lies partially, or totally, outside of

the world of Mathematics, it has to be dealt with at least some terms

that are not part of the mathematical lingo.

With the intervals description and the operations, the problem has to

be that it is unacceptable to equate ∞ to 5. We cannot, obviously, use




∞ and 5 in the same mathematical situations, for they are entities of

different nature. ∞ is not a location in the real numbers axis, for

instance, but 5 is.

We obviously have to find alternative ways, which will probably be

more complex than the ways we use this far, of describing our ideas

in the Mathematics sceneries.

To mention one mathematically sound alternative, instead of writing

[5,∞), we should probably be writing {x є R/ 5 ≤ x}.

Once the intervals, in Mathematics, seem to have been inspired by

the Cartesian Plane representation, we should need to fix the

Cartesian Plane in order to finish with the problem involving infinity

and the intervals in Mathematics.

Notice that when we get |0 |1 in the Cartesian Plane, we shorten it

to [0,1], for instance.

When we get |0 >, we shorten it to [0,∞) instead.

The representation [1,∞) should be a translation of the idea {x є R/ 1

≤ x}, but it is, as we have already proven in this paper, another

statement instead: It actually means

|1 |∞.

We do not write |1 |∞ because we do not believe that it is possible to fit the length

in our piece of paper (we write |1 > instead).

In order to translate the idea {x є R/ 1 ≤ x} into an interval, we have,

therefore, to create another symbol, a symbol that does not create




confusion, as explained before in this very paper.

Once more, writing things in a different way: Because using the

Cartesian Plane as a mathematical tool would lead us to, amongst

other things, believe that [1,∞) is a viable option, the Cartesian Plane

must be replaced with another space.

This special space, beyond Mathematics, or simply outside of it, would

have to satisfy the identified psychological need of mathematicians of

seeing things from a geometric perspective and should not imply any

mistake in the Mathematics through subliminal psycholinguistic

processes.

As a first requirement, this special space, to allow us to state that we

are presenting something more scientifically bearable than the

Cartesian Plane, must provide us with a way of stating clearly

(therefore also geometrically) that the referent infinity does not

belong to the world of Mathematics; for infinity belongs to somewhere

connected, of course, but somewhere of different nature to that of the

world of Mathematics, that is, to what we could call

EXTRAMATHEMATICS (see [Bunyan 1997], for instance).

As an illustrative allurement, Extramathematics could be told to be to

Mathematics the same thing that Metaphysics is for human kind: A

sort of connection between what is of its nature and what is beyond.

Basically, the own numbers, poor things, will never get to infinity if

keeping their nature, the same way humans will never get to the

Infinis que seras tamen10/24



spiritual world if doing so: Trascendence is a necessity in both cases,

that is, acquiring temporarily a nature that is different from the usual

nature, that goes beyond.

It is as if God and infinity did not exist in reality (reality = usual

nature's world?), what existed were everything in the middle, but, if

we did not invent God and infinity, we would not have an

anthropocentric world and we then could think that progressing in the

direction of the absolute knowledge or the absolute control over all

that there is is an impossible task, what could lead us to stop

investing in Science, for instance.

Also, to the side of why, human beings apparently love either being

bossed, commanded, adoring things and people, or thinking that they

will be able to occupy the position of whatever, or whoever, to them is

superior some day. If the figure of the boss did not exist, the vast

majority of the human beings apparently would feel lost, in a world

with no leaders to follow, that is, in a world with nobody to be blamed

for their mistakes who is not themselves, what could also lead to

absence of motivation to invest in their human existence, for

instance.

To insert all that we have placed in the EXTRAMATHEMATICS world in

the Mathematics world once more, we will make use of our World of

Infinita, where nothing ever reaches, mainly because it does not exist

but we imagine that it does, and we will work with the infinita the




same way we work with the imaginary numbers, for the square root of

a negative number exists as much as the physical place that no

number ever reaches does.

3. World of Infinita

Apparently, both mathematicians and logicians, so far, have never

really acknowledged the enormous differences between Language

and Mathematics, perhaps not even as school disciplines.

For instance, if one utters, in the English language, I will go to

infinity!; is that right or wrong?

The answer is simple: There is no right or wrong here, for a person

may utter whatever they like, sensible or not; there are no rules for

utterances. There are fixed rules for punctuation, for spelling, and

etc., but not for what may, or may be not, uttered.

They say I will go to infinity! and the message that they wish to

convey is sometimes even fully understood by the intended recipient.

Probably something similar to I will go to nowhere where you can find

me.

Mathematics, however, whilst part of Science, must hold severely tied

discourse, which allow for abstraction over abstraction to develop in

the direction of the true progress. This way, we should always worry

about the study of its terms, making sure that no mistakes occur in its




lingo, specially inconsistencies.

In human language, the idea of place, when the term infinity is used,

is unavoidable. To mention one example, Look at the infinity! brings

the term infinity associated with the idea of skies, therefore place.

We also say myriad, or infinity, of coins (means uncountably many in

our heads but, for Mathematics, the infinity of coins in this sentence

actually means countably many, for the number of coins produced on

Earth is always finite) and the idea that is intended by this sentence is

that of a subject getting exhausted of trying to even think about how

to count the coins, so that the sensation of the person uttering this

sentence is that of the amount of coins comparing to the size of the

sky, a place: A place of fatigue, fatigue that relates to the repetitive

unsuccessful trials of fully controlling things (from a human

perspective).

Some people seem to have extended the same allowances of the

human language to the mathematical lingo but whilst, in language,

almost anything can be accepted, perhaps under special conditions at

least sometimes; in Mathematics, very little can be accepted, the

most limiting condition of the mathematical lingo being that every

term in it must be univocal.

Those people have wrongly called infinity both the myriad of numbers

around a specific number in the real axis and the place that is not

determined, is never reached in reality by both eyes and imagination.




The two ideas are substantially different in nature and, therefore,

could only increase mistake, and probability of mistake, in Science if

taken to be the same, that is, if being assigned the same referent in

the mathematical lingo.

Each term of the mathematical lingo should always, if possible, satisfy

a condition that we could call perfect bijection, that is, should

univocally point to a certain object (of either concrete or abstract

nature) and have the pointed object univocally pointing to it in return.

The idea that comes easily to our minds, after assessing the just

exposed issues, is that we should drop the referent infinity and come

up with other names, names that are not common in language,

created for the specific end of pointing to the world references that

were previously designated by the term infinity. That is obviously the

only way to go, once the referent infinity is part of the usual language

and is not univocal, what makes the term useless for mathematical

purposes, as explained before.

Proceeding this way, we start with taking the smallest piece we may

physically, and easily, get in a ruler to mean X . We then say that X is

a unit of measurement for the ruler. Next, we call each unit infinitum.

Then, the set of all infinita will form the whole ruler, or axis. With this,

instead of referring to infinity as a boundary, we will refer to the

supremum of the infinita, or the infimum of the minus infinita; both

concepts with no reference either in the ruler or in the world we live




-empty referents (or sigmatoids), but at least expressing the right

idea in the Mathematics.

Our theory also solves the problems pointed by Parr, the issues

involving Aleph: size of infinity, types of infinity, and etc.

Now, one immediately understands how large the axis is when

compared to its pieces, what is simply logical and makes it all

coherent: Small pieces are infinitum, larger pieces are infinita, and the

whole lot is the interval between the infimum of the negative infinita

and the supremum of the positive ones.

We then understand that 0.5 in the axis actually means 0.5 infinitum,

what immediately makes us associate that with uncountably many

units contained in it, what is definitely different from considering half

a unit only (like half a chocolate?).

From now onwards, we write x infinita meaning simply x , the real

number, and such a writing will help us understand and deal with the

physical universe, once it helps us keep in the mind the difference

between what belongs to Mathematics and what belongs to Physics,

what is trivially necessary.

What follows is that it will be square infinita, for instance, while things

are in the context of Mathematics, instead of square meters. This

dissociates Mathematics from reality of things, making it all more

sensible, for the world of Mathematics will never fit reality with 100%

accuracy; it will fit this way only the world of an abstract ruler (for the




concrete ruler usually bears mistakes of at least physical order) or the

world of a mathematical object in general. This is a necessary

dissociation, precisely to reduce the amount of vain discussions,

unsolvable problems, and paradoxes in Science (those will be stopped

by the time of generation now, as a consequence, at least those

relating to the issues we deal with here).

We then suggest that other elements, besides axe and title, become

essential part of every mathematical graph of the Cartesian type.

These elements form a triplet: size of the infinitum, origin, and time3.

These elements should always respect the given order, so that

anyone who read the graph know what the graph is about.

Notice that, so far, the Cartesian Plane did not have rules for placing

explicit elements from the domain, or from the image, of a function

inside of it. Yet, the Cartesian Plane is one of the most important

mathematical entities ever created and everything else in

3 The time that we refer to here is the time of creation of the graph or of collection

of the data, the latter being preferred, if available, to the former. Notice what a

difference it all makes, in terms of mathematical elements, when it comes to

analyzing graphs and understanding them: Now Mathematics will acknowledge the

relevance of time, for just one second more could mean a centimeter more for

element of domain 5, for instance, x=5. Now, graphs are finally also what

Mathematics is about: Static pictures of a reality that has already passed and is fully

under control (for further details on how important this little modification in the

specifications of graphs in Mathematics is, see our work on Russell's Paradox ([M. R.

Pinheiro 2008])).




Mathematics is found completely tied to rules.

We are, therefore, starting to axiomatize this piece of Mathematics,

once every mathematical object, if truly mathematical, must be born

of some sort of axiom of definition.

It is worth mentioning that our choice of infinitum size should be

logical.

To mention an example, we may get a graph where the domain

elements differ by 0.5, the lowest-in-value domain element is 1, and

the image elements differ by 0.2. The person thinks that they are

ready to draw the graph by 2 pm of the 15th of June of 2011. In this

case, we may elect 0.5 our infinitum size in the horizontal axis

(smallest necessary slice of the domain) and 0.2 our infinitum size in

the vertical axis. Because the maximum detail of the data is one

decimal place, we'd better choose this format for every spelled

member of the axis. On the other hand, it may be better placing the

origin at (1;0). We would need all the details about the data to make

sure that our choice is the logical choice, but we do not have the

image of the points for our example. Assuming that this is all we

know, our graph, to follow our rules, should bring the triplet ((0.5;0.2),

(1;0), 15 /06/2011 at 2 pm) soon after the title.

The difference between the World of Infinita and the Cartesian Plane

might be subtle, but it helps absolutely everyone involved at both

subconscious and psycholinguistic levels. Now, everyone is kept




aware of the presence of an infinite number of members in each non-

degenerated interval of the real line. On the other hand, with our

triplets, we also become aware of the difference between the

infinitum of size 0.5 and the infinitum of size 1.3, for instance, that is,

we understand that there are less elements in one than in the other.

Some changes will not be a big deal. For instance, all the operations

that used to return infinity in Mathematics so far will return one of the

options below instead:

a) Sup Infta;

b) Inf -Infta.

Besides, all the operations that used to contain infinity4 will now have,

instead, one of the elements above (a or b).

Worth remembering that the way infinity is currently found in Mathematics makes us

think that infinity is an actual number , passive of becoming a member of divisions,

multiplications, and etc. However, this is incompatible with the current knowledge of

the meaning of the entity created by John Wallis in 1655.

With our notation, we now have something that is, in Mathematics, passive of proper

4 Originally, the symbol used for infinity was found replacing a determined numeral

in the sequence of the Roman numerals (see, for instance, [Weisstein 1999]).

Notwithstanding, in Mathematics, we cannot accept symbols, or words, for its lingo that have already

been used somewhere else to mean something else in the own Mathematics, for everything in

Mathematics should be univocal. Therefore, we are actually fixing more historical mistakes than the ones

we have initially waved with by producing this new notation, or these new referents, for the mathematical

lingo.




inclusion in a division, in a multiplication, or in any other operation, as a member.

Even the immediate understanding of what is going on in the operation of division

between 1 and x, for instance, using the example from [Weisstein 1999], when x goes to

what should be the largest real number available, is now incredibly improved, for all is

consistent and coherent with the pre-existent mathematical concepts that form a

requisite, in terms of learning, for us to be able to understand what goes on in the

operation calculating the limit of .

While, before we change things, a teacher would have to teach infinity soon after

introducing the concept of limit, now they may simply add the concept of limit instead,

following all other mathematical concepts, so that things do not look as if they are being

manufactured , forced , or forged .

4. Practice: One example.

We now present one example of a graphical situation in which, previously, we would be

including the Cartesian Plane as environment of display to show how easy it is to

replace the Cartesian Plane with the World of Infinita.

From [Dendane 2008]:

Problem 5: The cost of producing x tools by a company is given by

C(x) = 1200 x + 5500 (in $)

a) What is the cost of 100 tools?b) What is the cost of 101 tools?c) Find the difference between the cost of 101 and 100 tools.d) Find the slope of the graph of C?e) Interpret the slope.




To draw the graph describing the problem, the best thing to do is dividing the given

function by 1000. In this case, each unit of the previous function will be replaced with

one thousandth of it in the graph.

Once the main intents of the problem are seeing the slope of the graph of the function,

we should choose at least two domain numbers to work with.

Suppose that we choose x = 0 and x = 1, attaining C 10=5.5 and C 1

1=6.7 .

Going beyond the need (calculating the slope), we also choose x = 2 and form one more

ordered pair: (2; 7.9).

This way, our infinitum could be worth 1 for our horizontal axis and 1.2 for our vertical

axis. The time of collection of the data may be assumed to be today and now, for it is

not relevant in the problem proposal or we cannot investigate the origin of the data. The

origin of the system may be the usual one: (0;0). This way, we would end up with the

following graphical description for our situation:

World of Infinitum or W. I.

((1 tool; 1.2 dollars (original cost ÷ 1000)), (0;0), 15th June 2011 at 3:00 pm)

C. D. Infta

D. Infta




In the World of Infinita, we call our Cartesian look-alike axes Domain of Infinita and

Counter-Domain of Infinita5

, instead of Domain and Counter-Domain, which are

usually referred to as x and y, and represent their names by means of acronyms.

Were our W. I. displaying the graph that describes the Problem 5 in full, our horizontal

axis would show the numbers 0, 1, and 2 and our vertical axis would show the numbers

5.5, 6.7, and 7.9.

The complete graph for the Problem 5 should read, in the W.I.:

Cost per tool, company unknown, slope problem

((1 tool; 1.2 dollars (original cost ÷ 1000)), (0;0), 15th June 2011 at 3:00 pm)

7.9 C. D. Infta

6.7

5.5

D. Infta

0 1 2

Notice that the person dealing with the representation of a situation in the W. I. will

have no difficulties in understanding that because the numbers hold clear separation, or

5 We do not call the horizontal axis tools and the vertical axis cost because the axe cannot be called that

way. Basically, if we do such, we will end up with a slope that is 1.2 dollars/tool , rather than 1.2, what

is unacceptable in the Mathematics (remember that the slope is the tangent of the angle in which the

line lies over the horizontal axis).




distinction, amongst themselves, and the physical objects, described by them in the

graphs, are continuous in nature, or because the physical entities are not passive of

fractioning beyond a certain level and the mathematical entities are, there is a huge

distinction between the physical and the mathematical universe, which can only be dealt

with through an artificial, highly manufactured, and exotic, world, such as the World of

Infinita that we here propose.

5. Conclusion:

In this paper, we have proposed the World of Infinita as a replacement

for the Cartesian Plane.

The World of Infinita (W.I.) is, finally, a proper mathematical tool, once

its use does not generate mistakes in the Mathematics.

We have also proposed that we abandon both the symbol and the

referent infinity in the Mathematics, once we prove that they both

constitute mistaken conceptions when the foundations of both

Mathematics and Language are considered.

To replace the referent infinity , we have created the referents

infinitum and infinita and we have made them be represented by

abbreviations plus existent mathematical symbols.




6. References:

[Bunyan 1997] A. Bunyan, T. Benwell, M. Errey, et al. Prefixes. (1997).

Englishclub.com. http://www.englishclub.com/vocabulary/prefixes.htm. Accessed on

the 15th of July of 2009.

[Dendane 2008] A. Dendane. Linear function problems with solutions.

(2008). Analyzemath.

http://www.analyzemath.com/math_problems/linear_func_problems.ht

ml. Accessed on the 17th of July of 2009.

[H. C. Parr 2003] H. C. Parr. Infinity. (2003). Available online at

http://www.cparr.freeserve.co.uk/hcp/Infinity.htm. Accessed on the 3rd

of May of 2008.

[Merriam-Webster-1 2009] Merriam-Webster's authors. Infinity. (2009). In Merriam-

Webster Online Dictionary. http://www.merriam-webster.com/dictionary/Infinity.

Accessed on the 15th of July of 2009.

[M. R. Pinheiro 2008] M. R. Pinheiro. Completeness and consistency of

Arithmetics. (2008). Preprint located at scribd, illmrpinheiro and

illmrpinheiro2. Accessed on the 1st of August of 2009.


http://www.englishclub.com/vocabulary/prefixes.htm


http://www.analyzemath.com/math_problems/linear_func_problems.html


http://www.cparr.freeserve.co.uk/hcp/Infinity.htm

http://www.merriam-webster.com/dictionary/infinity





http://www.cparr.freeserve.co.uk/hcp/Infinity.htm




[S. Schwartzman 1994] S. Schwartzman. The Words of Mathematics:

An Etymological Dictionary of Mathematical Terms. MAA.

ISBN:0883855119.

[Weisstein 1999] E. W. Weisstein. Infinity. (1999). MathWorld.

http://mathworld.wolfram.com/Infinity.html . Accessed on the 16th of July of 2009.


http://mathworld.wolfram.com/Infinity.html






infinis que sera tamen

Documents