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Abstract

We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M \geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic $\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is weakly projective with respect to the real Diophantine arithmetic $\mathbf{R}=(\mathbb{R},+,\times,\leq_{\mathbb{R}})$, while the third one is projective with respect to the extended real Diophantine arithmetic $\overline{\mathbf{R}}=(\overline{\mathbb{R}},+,\times,\leq_{\overline{\mathbb{R}}})$. In addition, we have that every $\mathbf{A}_M$ and every $\mathbf{B}_M$ are a complete totally ordered semiring, while every $\mathbf{A}_{-M,M}$ is not. We show that the projection of any series of elements of $\mathbb{R}_+$ converges in $\mathbf{A}_M$, for any $M \geq 1$, and that the projection of any non-oscillating series series of elements of $\mathbb{R}$ converges in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M > 0$. We also prove that working in $\mathbf{A}_M$ and in $\mathbf{A}_{-M,M}$, for any $M \geq 1$, and in $\mathbf{B}_M$, for all $M>0$, allows to overcome a version of the paradox of the heap.
CONCERNING THREE CLASSES OF NON-DIOPHANTINE
ARITHMETICS
MICHELE CAPRIO, ANDREA AVENI AND SAYAN MUKHERJEE
Abstract. We present three classes of abstract prearithmetics, tAMuMě1,tA´M,M uMě1,
and tBMuMą0. The first one is weakly projective with respect to the nonnegative real
Diophantine arithmetic R`“ pR`,`,ˆ,ďR`q, the second one is weakly projective with
respect to the real Diophantine arithmetic R“ pR,`,ˆ,ďRq, while the third one is exactly
projective with respect to the extended real Diophantine arithmetic R“ pR,`,ˆ,ďRq. In
addition, we have that every AMand every BMare a complete totally ordered semiring,
while every A´M,M is not. We show that the projection of any series of elements of R`
converges in AM, for any Mě1, and that the projection of any non-indeterminate series
of elements of Rconverges in A´M,M , for any Mě1, and in BM, for all Mą0. We also
prove that working in AMand in A´M,M , for any Mě1, and in BM, for all Mą0, allows
to overcome a version of the paradox of the heap.
1. Introduction.
Although the conventional arithmetic – which we call Diophantine from Diophantus, the
Greek mathematician who first approached this branch of mathematics – is almost as old as
mathematics itself, it sometimes fails to correctly describe natural phenomena. For example,
in [4] Helmoltz points out that adding one raindrop to another one leaves us with one rain-
drop, while in [13] Kline notices that Diophantine arithmetic fails to correctly describe the
result of combining gases or liquids by volume. Indeed, one quarter of alcohol and one quar-
ter of water only yield about 1.8 quarters of vodka. To overcome this issue, scholars started
developing inconsistent arithmetics, that is, arithmetics for which one or more Peano axioms
were at the same time true and false. The most striking one was ultraintuitionism, developed
by Yesenin-Volpin in [14], that asserted that only a finite quantity of natural numbers exists.
Other authors suggested that numbers are finite (see e.g. [2] and [5]), while different scholars
adopted a more moderate approach. The inconsistency of these alternative arithmetics lies
in the fact that they are all grounded in the ordinary Diophantine arithmetic. The first
consistent alternative to Diophantine arithmetic was proposed by Burgin [7], and the name
non-Diophantine seemed perfectly suited for this arithmetic. Non-Diophantine arithmetics
for natural and whole numbers have been studied by Burgin in [7,8,9,10], while those for
real and complex numbers by Czachor in [1,6]. A complete account on non-Diophantine
arithmetics can be found in the recent book by Burgin and Czachor [11]. There, for exam-
ple, the authors show how non-Diophantine arithmetics are crucial for the development and
application of different kinds of non-Newtonian calculi [3,6,12].
2010 Mathematics Subject Classification. Primary: 03H15; Secondary: 03C62.
Key words and phrases. Non-Diophantine arithmetics; convergence of series; paradox of the heap.
2M. Caprio, A. Aveni and S. Mukherjee
There are two types of non-Diophantine arithmetics: dual and projective. In this paper,
we work with the latter. We start by defining an abstract prearithmetic A,
A:“ pA, `A,ˆA,ďAq,
where AĂRis the carrier of A(that is, the set of the elements of A), ďAis a partial
order on A, and `Aand ˆAare two binary operations defined on the elements of A. We
conventionally call them addition and multiplication, but that can be any generic operation.
Naturally, the conventional Diophantine arithmetic R“ pR,`,ˆ,ďRqof real numbers is an
abstract preartithmetic.1
Abstract prearithmetic Ais called weakly projective with respect to a second abstract
prearithmetic B“ pB, `B,ˆB,ďBqif there exist two functions g:AÑBand h:BÑA
such that, for all a1, a2PA,
a1`Aa2hpgpa1q `Bgpa2qq and a1ˆAa2hpgpa1q ˆBgpa2qq.
Function gis called the projector and function his called the coprojector for the pair pA,Bq.
The weak projection of the sum b1`Bb2of two elements of Bonto Ais defined as hpb1`Bb2q,
while the weak projection of the product b1ˆBb2of two elements of Bonto Ais defined as
hpb1ˆBb2q.
Abstract prearithmetic Ais called exactly projective with respect to abstract prearithmetic
Bif it is weakly projective with respect to B, with projector f´1and coprojector f. We call
f, that has to be bijective, the generator of projector and coprojector.
Weakly projective prearithmetics depend on two functional parameters, gand h– one, f,
if they are exactly projective – and recover the conventional Diophantine arithmetic when
these functions are the identity. To this extent, we can consider non-Diophantine arithmetics
as a generalization of the Diophantine one.
In this work, we consider three classes of abstract prearithmetics, tAMuMě1,tA´M,M uMě1,
and tBMuMą0. They are useful to describe some natural and tech phenomena for which the
conventional Diophantine arithmetic fails, and their elements allow us to overcome the version
of the paradox of the heap (or sorites paradox) stated in [10, Section 2]. The setting of this
variant of the sorites paradox is adding one grain of sand to a heap of sand, and the question
is, once a grain is added, whether the heap is still the same heap. The heart of sorites paradox
is the issue of vagueness, in this case vagueness of the word “heap”.
We show that every element AMof the first class is a complete totally ordered semiring, and
it is weakly projective with respect to R`. Furthermore, we prove that the weak projection
of any series řnanof elements of R`:“ r0,8q is convergent in each AM.
The elements A´M,M of the second class allow to overcome the paradox of the heap and
are weakly projective with respect to the real Diophantine arithmetic R“ pR,`,ˆ,ďRq.
The weak projection of any non-indeterminate series řnanof terms in Ris convergent in
A´M1,M1, for all M1ě1. The drawback of working with this class is that its elements are
not semirings, because the addition operation is not associative.
The elements of tBMuMą0too can be used to solve the paradox of the heap. They are com-
plete totally ordered semirings, and are exactly projective with respect to the extended real
1Notice that `is the usual addition, ˆis the usual multiplication, and ďRis the usual partial order on R.
On three classes of Non-Diophantine arithmetics 3
Diophantine arithmetic R“ pR,`,ˆ,ďRq. The exact projection of any non-indeterminate
series řnanof terms in Ris convergent in BM1, for all M1ą0.
AMA´M,M BM
AM“ r0, M sA´M,M “ r´M, M sBM“ r0, Ms
Solves paradox of the heap Solves paradox of the heap Solves paradox of the heap
Weakly projective wrt R`Weakly projective wrt RExactly projective wrt R
Complete totally ordered semiring Not a semiring Complete totally ordered semiring
Weak projection of a series of elements of
R`is convergent
Weak projection of a series of elements of
Ris absolutely convergent
Exact projection of a series of elements of
Ris absolutely convergent
Weak projection of a non-indeterminate series
of elements of Ris convergent
Exact projection of a non-indeterminate series
of elements of Ris convergent
Table 1. A summary of the properties of the classes we introduce in this
paper. All of them can be used to describe those (natural and tech) phenomena
for which Diophantine arithmetics fail.
The paper is divided as follows. Section 2deals with tAMuMě1. Section 3presents the
class tA´M,M uMě1, while tBMuMą0is discussed in Section 4.
2. Class tAMuMě1.
For any real Mě1, we define the corresponding non-Diophantine preartithmetic as
AM“ pAM,,b,ďAMq
having the following properties:
(i) The order relation ďAMis the restriction to AMof the usual order on the reals;
(ii) AMĂR`has maximal element Mand minimal element 0with respect to ďAM, and
is such that
0PAM, which ensures having a multiplicative absorbing and additive neutral
element in our set;
1PAM, which ensures having a multiplicative neutral element in our set;
there is at least an element xP p0,1qsuch that xPAM;
(iii) It is closed under the following two operations
:AMˆAMÑAM,:pa, bq ÞÑ ab:minpM, a `bq
and
b:AMˆAMÑAM,b:pa, bq ÞÑ abb:minpM, a ˆbq,
where `and ˆdenote the usual sum and product in R, respectively.
Proposition 1. Addition is associative.
Proof. Pick any a, b, c PAM. Let abd, where da`bif a`bďMor dMif
a`bąM. Let also bce, where eb`cif b`cďMor eMif b`cąM. Then,
pabq ‘ cdc#d`c“ pa`bq ` cif d`cďM
Mif d`cąM
4M. Caprio, A. Aveni and S. Mukherjee
and
a‘ pbcq “ ae#a`ea` pb`cqif a`eďM
Mif a`eąM.
But we know that pa`bq ` ca` pb`cq, so the result follows.
Since addition is associative we have that, for every kPN,
k
à
n1
xnmin ˜M,
n
ÿ
n1
xn¸.
By imposing on Mthe relative topology derived from R, we can define
8
à
n1
xn:lim
kÑ8
k
à
n1
xnmin ˜M,
8
ÿ
n1
xn¸.
Proposition 2. AM“ r0, M s.
Proof. Recall that AMis closed under products band arbitrary summations . Fix any
xP p0,1qand let zP p0, Ms. Then, it is always possible to express
z
8
à
n1˜kn
à
j1
xn¸,
for some pknq P NN, defined recursively as follows
k1maxtmPN:ząmxu;
kj`1maxtmPN:ząřj
u1kuxu`mxju.
It is clear that the partial sums are always less than zďM, and so are all nonnegative and
well defined. Moreover,
ˇˇˇˇˇ
z´
m
à
n1˜kn
à
j1
xn¸ˇˇˇˇˇ
ďxmÑ0.
Finally observe that the result of our two operations is always less or equal than Mand,
from non-negative numbers, it is impossible to obtain negative numbers.
This result implies that AMis a complete totally ordered semiring. In this Section, we
consider the class tAMuMě1of abstract prearithmetics.
Remark. Notice that AMcannot be a ring because for any aPAMzt0u, it lacks the additive
inverse ´a; this because we defined AMto be a subset of R`. Notice also that in this abstract
prearithmetic Mis an idempotent element, that is, MM‘ ¨ ¨ ¨ MM.
2.1. Overcoming the paradox of the heap. The paradox of the heap is a paradox that
arises from vague predicates. A formulation of such paradox (also called the sorites paradox,
from the Greek word σωρoς , “heap”), given in [10, Section 2], is the following.
(1) One million grains of sand make a heap;
(2) If one grain of sand is added to this heap, the heap stays the same;
(3) However, when we add 1to any natural number, we always get a new number.
On three classes of Non-Diophantine arithmetics 5
This formulation of the paradox of the heap is proposed by Burgin to inspect whether adding
$1to the assets of a millionaire makes them “more of a millionaire”, or leaves their fortune
unchanged. We use the class tAMuMě1to address paradox of the heap. Indeed, it is enough
to take the element of the class for which M1000000, so that when we perform the addition
M1, we get M. This conveys the idea that adding a grain of sand to the heap leaves us
with a heap.
The abstract prearithmetic we introduced can also be used to describe phenomena like the
one noted by Helmholtz in [4]: adding one raindrop to another one gives one raindrop, or the
one pointed out by Lebesgue (cf. [13]): putting a lion and a rabbit in a cage, one will not
find two animals in the cage later on. In both these cases, it suffices to consider the element
of the class for which M1, so that 111.
AMallows us also to avoid introducing inconsistent Diophantine arithmetics, that is, arith-
metics for which one or more Peano axioms were at the same time true and false. For example,
in [2] Rosinger points out that electronic digital computers, when operating on the integers,
act according to the usual Peano axioms for Nplus an extra ad-hoc axiom, called the machine
infinity axiom. The machine infinity axiom states that there exists ˘
MPNfar greater than 1
such that ˘
M`1˘
M(for example, ˘
M231 ´1is the maximum positive value for a 32-bit
signed binary integer in computing). Clearly, Peano axioms and the machine infinity axiom
together give rise to an inconsistency, which can be easily avoided by working with A˘
M.
In [5], Van Bendegem developed an inconsistent axiomatic arithmetic similar to the “ma-
chine” one described in [2]. He changed the Peano axioms so that a number that is the
successor of itself exists. In particular, the fifth Peano axiom states that if xy, then xand
yare the same number. In the system of Van Bendegem, starting from some number n, all
its successors will be equal to n. Then, the statement nn`1is considered as both true
and false at the same time, giving rise to an inconsistency. It is immediate to see how this
inconsistency can be overcome by working with any abstract prearithmetic AMin our class.
2.2. AMis weakly projective with respect to R`.Pick any AM1P tAMu, and consider
R`“ pR`,`,ˆ,ďR`q. Consider then the functions:
g:AMÑR`,aÞÑ gpaq ” a, so gis the identity function Id|AM;
h:R`ÑAM,aÞÑ hpaq:minpM, aq.
Now, if we compute hpgpaq ` gpbqq, for all a, b PAM, we have that
hpgpaq ` gpbqq “ hpa`bq “ minpM, a `bq “ ab.
Similarly, we show that hpgpaq ˆ gpbqq “ abb. Hence, addition and multiplication in R`are
weakly projected onto addition and multiplication in AM1, respectively. So, we can conclude
that AM1is weakly projective with respect to R`, for all M1ě1.
2.3. Series of elements of AM.Consider any series Ànanof elements of AM. Its corre-
sponding series řngpanqin R`can only be convergent or divergent to `8. It cannot be
divergent to ´8 because we are summing positive elements only, and it cannot be neither
convergent nor divergent (i.e. it cannot be indeterminate), because the elements of the series
cannot alternate their sign. But since AMhas a maximal and a minimal element, Mand 0
respectively, this means that Ànanis always convergent.
6M. Caprio, A. Aveni and S. Mukherjee
2.4. Weak projection of series in R`onto AM.In this Section, we show that the weak
projection of any series of elements of R`converges in AM, for all Mě1. This is an exciting
result because it allows the scholar that needs a particular series to converge in their analysis
to reach that result by performing a weak projection of the series onto AM, and then continue
the analysis in AM.
Consider any series řnanof elements of R`. Similarly to what we pointed out before,
it can be convergent or divergent to `8. It cannot be divergent to ´8 because we are
summing positive elements only, and it cannot be neither convergent nor divergent (i.e. it
cannot be indeterminate), because the elements of the series cannot alternate their sign. Let
us then show that the weak projection of ř8
n1an:lim
kÑ8 řk
n1anis convergent.
First, suppose that ř8
n1anLăM. Then, hpř8
n1anq “ hpLq “ L.
Then, let ř8
n1anLěM. We have that hpř8
n1anq “ hpLq “ M, where in both cases
the last equality comes from the definition of h.
Finally, suppose that ř8
n1an“ 8. Then,
h˜8
ÿ
n1
an¸h˜lim
kÑ8
k
ÿ
n1
an¸lim
kÑ8 h˜k
ÿ
n1
an¸M,
where the second equality comes from hbeing continuous, and the last equality comes from
the fact that function his constant once its argument is equal to M. To check that his
continuous, we just need to check that it is continuous at M; but this is immediate, since
limaÑM´hpaq “ MlimaÑM`hpaq.
The following lemma comes immediately from Proposition 1.
Lemma 1. For any series ř8
n1anof elements of R`, for any kPN,
h˜8
ÿ
n1
an¸h˜k
ÿ
n1
an¸h˜8
ÿ
nk`1
an¸.
3. Class tA´M,M uMě1
For any real Mě1, we define the corresponding non-Diophantine prearithmetic as
A´M,M “ pA´M ,M ,,b,ďA´M,M q
having the following properties:
(i) The order relation ďA´M,M is the restriction to A´M ,M of the usual order on the reals;
(ii) A´M,M ĂRhas maximal element Mand minimal element ´Mwith respect to
ďA´M,M , and is such that
0PA´M,M , which ensures having a multiplicative absorbing and additive neutral
element in our set;
1PA´M,M , which ensures having a multiplicative neutral element in our set;
´Mă0ăMand there is at least an element xP p0,1qsuch that xPA´M,M ;
bPA´M,M ùñ ´bPA´M,M ;
On three classes of Non-Diophantine arithmetics 7
(iii) It is closed under the following two operations
:A´M,M ˆA´M,M ÑA´M,M ,
pa, bq ÞÑ ab:$
&
%
a`bif a`bP r´M, M s
Mif a`bąM
´Mif a`bă ´M
,
where `denotes the usual sum in R, and
b:A´M,M ˆA´M,M ÑA´M,M ,
pa, bq ÞÑ abb:$
&
%
aˆbif aˆbP r´M, M s
Mif aˆbąM
´Mif aˆbă ´M
,
where ˆdenotes the usual product in R.
Since A´M,M is closed under products band arbitrary summations , it follows, along
the lines of Proposition 2, that A´M,M “ r´M, M s, for all Mě1. Notice also that from
the definition of addition it follows immediately that is commutative, but A´M,M
pA´M,M ,,b,ďA´M,M qis not a semiring. Indeed, addition is not associative. An easy
counterexample is the following: ´2pM1q “ ´2MM´2, while 2Mq1
´1MM´1.
Despite that, the elements of tA´M,M uMě1can still be useful. Any A´M1,M1still solves
the paradox of the heap. If we then consider the functions
s:A´M,M ÑR,aÞÑ spaq ” a, so sis the identity function Id|A´M,M ;
t:RÑA´M,M ,aÞÑ tpaq:$
&
%
aif aP r´M, M s
Mif aąM
´Mif aă ´M
,
it is immediate to see that, for all M1ě1,A´M1,M1is weakly projective with respect to the
real arithmetic R“ pR,`,ˆ,ďRq, with projector sand coprojector t.
Proposition 3. Pick any real Mě1, and any sequence pxnq P RN. For every n, define
yn:řnďkxn. Then, the weak projection of pynqconverges in A´M,M if and only if one of
the following holds:
(i) lim infnyněM;
(ii) lim ynexists and belongs to M, Mq;
(iii) lim supnynď ´M.
Proof. The fact that tis continuous is obvious. Suppose then that limntpynq “ tplimnynq P
M, M q, where the equality comes from the continuity of t. By definition of t, we have
that either limnynbelongs to M, Mq, verifying condition (ii), or it does not. In this latter
case, it may be that either limnyněM, verifying condition (i), or limnynď ´M, verifying
condition (iii).
The fact that condition (ii) implies limntpynq P p´M, M qis immediate by the definition of
function t. Suppose now that lim infnyn“8ąM. This implies that limnyn“ 8. But then,
8M. Caprio, A. Aveni and S. Mukherjee
limntpynq “ tplimnynq “ M, where the first equality comes form the continuity of t, and the
last equality from the definition of t. A similar argument shows that if lim supnyn“ ´8,
then limntpynq “ ´M. If lim infnynLP rM, 8q, the previous argument shows that
limntpynq “ M, while if lim supnyn“ ´LP p´8,´Ms, the previous argument shows that
limntpynq “ ´M. This concludes the proof.
The following corollary comes immediately from Proposition 3.
Corollary 1. Pick any real Mě1. The weak projection of any series of elements of Ris
absolutely convergent in A´M,M . In addition, the weak projection of any series of elements
of Rthat is either convergent or divergent (to `8 or ´8), converges in A´M,M .
Notice that, because addition is not associative, Lemma 1does not hold in this class of
abstract prearithmetics. This means that there may exist a series ř8
n1anof elements of R
such that
t˜8
ÿ
n1
an¸t˜k
ÿ
n1
an¸t˜8
ÿ
nk`1
an¸.
Hence, we need to project the entire series in the exact order we want the elements to be
summed, otherwise Proposition 3may not hold.
There is a tradeoff in using this class instead of tAMu. We still manage to resolve the
paradox of the heap, and we further show that the weak projection of series that diverge
also to ´8, converges in A´M,M , for all Mě1. The shortcoming, however, is that we lose
associativity, so working with elements of A´M,M may be difficult. Ultimately, the choice of
one or the other will depend on the application the scholar has in mind.
4. Class tBMuMą0.
In this Section we present a class of abstract prearithmetics tBMuMą0where every element
is a complete totally ordered semiring, and such that the projection of a convergent or
divergent series (to `8 or ´8) of elements of Rconverges. Its elements can be used to solve
the paradox of the heap.
For every real Mą0, we define the corresponding non-Diophantine prearithmetic as
BM“ pBM,`,ˆ,ďBMq
having the following properties:
(i) The order relation ďBMis the restriction to BMof the usual order on the reals;
(ii) BM“ r0, Ms;
(iii) Let R:“ r´8,8s and consider the function
f:RÑBM, x ÞÑ fpxq:$
&
%
M´arctanpxq
π`1
2¯if xPR
Mif x“ 8
0if x“ ´8
On three classes of Non-Diophantine arithmetics 9
and its inverse
f´1:BMÑR, x ÞÑ f´1pxq:$
&
%
tan `π
M`x´M
2˘˘ if xP p0, Mq
8if xM
´8 if x0
.
Then, BMis closed under the following two operations
`:BMˆBMÑBM,pa, bq ÞÑ a`b:f`f´1paq ` f´1pbq˘,
where `denotes the sum in R, and
ˆ:BMˆBMÑBM,pa, bq ÞÑ aˆb:f`f´1paq ˆ f´1pbq˘,
where ˆdenotes the product in R.
Notice that we do not “force” 0and Mto be the boundary elements of BM; they come
naturally from the way addition `and multiplication ˆare defined. In addition, we have
that by construction BMis exactly projective with respect to R“ pR,`,ˆ,ďRq, and that its
generator induces an homeomorphism between Rand r0, Ms. This tells us immediately that
pBM,`,ˆ,ďBMqis a complete totally ordered semiring, so addition `and multiplication ˆ
are associative. The fact that `8 and ´8 in Rcorrespond to Mand 0, respectively, tells
us that the exact projection fpřnanqof any series řnanof elements of Rconverges in BM,
as long as řnanis not indeterminate. The elements of BMcan be used to solve the paradox
of the heap; to see this, notice that M`aM, for all aPBMand all Mą0. Also, Mis
an idempotent element of BM, for all Mą0:M`M`¨ ¨ ¨ `MM.
Acknowledgements
We are particularly grateful to two anonymous referees for their insightful comments and
suggestions.
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Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC
27708-0251
Email address:michele.caprio@duke.edu
URL:https://mc6034.wixsite.com/caprio
Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, NC
27708-0251
Email address:andrea.aveni@duke.edu
URL:https://www.researchgate.net/profile/Andrea_Aveni
Department of Statistical Science, Mathematics, Computer Science, and Biostatistics &
Bioinformatics, Duke University, Durham, NC 27708-0251
Email address:sayan@stat.duke.edu
URL:https://sayanmuk.github.io/
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Elemer E. Rosinger. On the Safe Use of Inconsistent Mathematics. Available at arXiv:0811.2405, 2008.
Non-Classical Models of Natural Numbers
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Mark Burgin. Non-Classical Models of Natural Numbers. Russian Mathematical Surveys, 32:209-210, 1977.
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Mark Burgin. Elements of Non-Diophantine Arithmetics. Proceedings of the 6th Annual International Conference on Statistics, Mathematics and Related Fields, pages 190-203, 2007.