1 Introduction

In this paper, we will examine two very burning issues of today’s learning disabilities, namely dyscalculia and dyslexia. More precisely, we will discuss their definitions and we will introduce a new concept in this area of research, namely the essence of mathematics as science, which—of course—is the vital point. This will lead us to the unequivocal conclusion (cf. Theorem 1, Remarks 5 and 8 below) that dyscalculia is not a concept by itself, but merely yet another one of the sad guises of dyslexia. This will be emphasized by the results of an inquiry made among students with these two learning disabilities, where these students have been asked the following questions.

  1. (i)

    In which part/parts of the sequence ’reading and understanding a mathematical problem/elaborating/fomulating/writing/presenting a solution’ do you experience—encounter with—difficulties.

  2. (ii)

    Have you ever been experiencing problems with time and/or orientating yourself in everyday life and/or remembering courses of events.

  3. (iii)

    Do you feel that presenting your solutions to a mathematical problem is simplified by using a computer or similar.

Here, the dyslectics will run into problem at the beginning and at the end of the sequence in question \(\,(i)\,\) above, while the student with dyscalculia would encounter with difficulties in the mid section. The answers obtained have all been of the following kinds.

  1. (i)

    ‘I can read, but sometimes the letters and the words are jumping around.’;

    ‘I can read, but sometimes I don’t understand what I read and because of that, I can’t solve the problem.’;

    ‘I can read digits, but sometimes it’s also jumping around a bit, and then I have to read the problem many times. When I have read the problem many times, then I understand and can solve the problem.’

  2. (ii)

    ‘No, I can see how things happen in the right order.’

  3. (iii)

    ‘Yes, it’s much easier to use a computer, because I don’t write very well, so I can see what I have written much better on the computer.’;

    ‘I can read the letters much easier on the computer.’;

    ‘Multiplication is easier on the computer, because then the numbers don’t jump around.’;

    ‘I don’t know why, but it’s much easier to read both letters and numbers on the computer.’; ‘I can solve problems in mathematics both better and easier on the computer.’

2 The Nature of the Language and the Sad Guise of Dyslexia

The language, spoken and in writing, and the body language (referred to as a concept and thus independent of any particular idiom) is something the human race—with our eminent intellect—have developed during eons of time in order to be able to exchange thoughts, ideas, opinions and—to some extent—feelings with one another, i.e., the language—spoken and written—is the human race’s existing and adequate means of communication. In this note, we choose to keep focus on the language spoken and written (henceforth referred to as—simply—the language). This is also, above all others, the academic world’s principal means of communication.

However, the language manifests significant shortcomings in potential to express mainly i.a. feelings. The language does not offer adequate means of, clear and unequivocally, expressing our inmost meanings and our inmost feelings, but these abstract phenomena every now and then evince significant discrepancies between the sequences experience/feel and dress in words/by means of writing or verbally impart to the world around us. In every sequence, there will always be some noise and some losses, and if I e.g. get an inner picture of my neighbour’s red car, and I utter the words my neighbour’s red car before a collection of ten people, these people will probably first of all paint a picture of a red car and then try to render their associations. This will most likely lead to ten different allusions to my neighbour’s red car, which was the original toe hold for take off for the entire discussion.

A dyslectic is (cf. Lundberg and Sterner 2009, p. 33; Specialpedagogiska Skolmyndigheten 2012) a person with reading—and writing difficulties but with an—in other parts—fine working intellect. In the process of experience/feel and dress in words/by means of writing or verbally impart to the world around us, the subsequence of experiencing, feeling and dress in words will not pose an obstacle. In stead, the difficulties will arise during the sequence of by means of writing impart to—or collect from—the world around us. Cf. the quotation from Sect. 1: ’I can read, but sometimes I don’t understand what I read and because of that, I can’t solve the problem.’ This strongly indicates that the crux of the matter does not lie in understanding, treating and elaborating a solution to a problem, but in writing conveying the solution to the reader. This is also underlined by the fact that most dyslectics are very benefitted by the use of a computer, where the shaping and distinction of different letters becomes a smaller issue.

3 The Concept of Dyscalculia

In Lundberg and Sterner (2009, pp. 20–24), (and further references therein) dyscalculia is defined as reduced ability of determining numbers and developing a mental number axis, but where, in other parts and quite analogously to a dyslectic, the person suffering from it has a well-behaved intellect. Though this definition could appear somewhat imprecise, it should be stressed that before the student has received the diagnosis dyscalculia, his or her arithmetic skills, and how these abilities have developed over time, have been subject for studies spanning a rather long period with many—more or less—conclusive tests together leading up to the diagnosis of dyscalculia, which also could be of different severity from person to person.

Remark 1

Note that this is the generally adopted definition of the most prime val form of dyscalculia—and hence the origin of all problems with arithmetics referrable to dyscalculia. It is having problems of distinguishing between different natural numbers, i.e., distinguishing between ’how many’, i.e., developing a mental number axis; but having an—in other parts—fine working intellect. The discrepancies of this definition is the outcome of Theorem 1, which proves that—in fact—dyscalculia in this form cannot exist.

Thus, dyscalculia in its primeval form—as commonly known—is difficulties in understanding the construction of the mathematical system of the natural numbers and developing a mental number axis, i.e., the set of numbers \({\mathbb {N}}=\{0,1,2,\ldots \}\) and its mathematical structure with the ordinary binary operations of addition \(+\) and multiplication \(\cdot \) (cf. also Peters et al. 2018; Szücs et al. 2013). In the process of elaborating a mental number axis it is natural to (cf. Lundberg and Sterner 2009, p. 20) logarithmically placing the numbers on the axis—the higher the numbers, the bigger the tendency of lumping them together—so that neighbouring numbers will be lying closer together than numbers at the beginning of the scale. Not until after a couple of years of formal education in mathematics will the scale become linear according to Lundberg and Sterner (2009).

The following examples are recited from Lundberg and Sterner (2009, pp. 20–21).

Example 1

This example (cf. Lonnemann et al. 2008) is presenting a test, where three numbers, e.g. 57, 64, 92, are shown to the student, and where the problem is to, as fast as possible, tell which distance, with respect to quantity, is the largest; the distance between 57 and 64, or the distance between 64 and 92. The physical distance between the three numbers is altered in a variety of ways, where, in one condition, a small distance between 57 and 64, and a large distance between 64 and 92 is marked

$$\begin{aligned} (57,...................64,.......................................................92). \end{aligned}$$

Here, we thus have congruence between the differences in size between the numbers and the physical distance between them. In another condition, there is incongruence:

$$\begin{aligned} (57,.......................................................64,...................92). \end{aligned}$$

Those with a mental, linear number axis apparently will be encountering with more difficulties in the incongruent condition, yielding longer reaction time and more errors, while the difficulties in the two conditions above will be of more equal degree for those, who have as yet not developed a functionable number axis, i.e., students with claimed reduced counting abilities do not encounter with the same amount of difficulties in the incongruent case.

Example 2

In another example, the character of the number axis is determined by letting the student, on a horizontal line, mark where a given number should be placed. The line can e.g. have the numbers 0 and 100 marked at the two endpoints. For each marking conducted by the student, the distance to the correct position is measured. Thereupon, the student’s markings are plotted in a diagram, and they are compared with the linear position, whereupon one can calculate how much of the variation, which is due to a linear function, and to what extent a logarithmic function can characterize the students’ markings. The more linear the function, the more mature the sense of numbers and the less the risk of difficulties with numbers and calculations (cf. Opfer and Siegler 2007).

Example 3

Another interesting attempt in pinpointing the absolute form of dyscalculia has been performed by Butterworth (2003). He has developed a computer based screening instrument, where he is trying to isolate the basic lacking in ability in telling numbers (’how many’) (core systems). In a partial test, the student is shown a picture of a number of filled circles or other subjects in the left part of the computer screen, while the right part shows a figure. The mission is simple. Which one of the filled circle or the figure does represent the largest number? The student answers by pressing one of two buttons, whereupon the lapse of time between the appearing of the picture and the pressing of the button is registered with the tolerance of 0.001 s. Butterworth testifies to that a correct answer alone is not sufficient, but the sense of numbers is inherent in the ability of rapidly discerning the correct number, when viewing the picture. Butterworth is of the opinion that a person with absolute dyscalculia is incapable of this.

For further details, cf. Lundberg and Sterner (2009). The common factor of all of these studies—and all the theories—is however the focus on the fact that people with dyscalculia are having difficulties with the structure of the aforementioned natural number system, where the starting point thus is that the natural numbers are totally connected with \(\,0,1,2,\ldots \,\) like we with our—for communication purposes—developed language have chosen to denote them. Although there certainly and undoubtedly exist people with genuine problems in handling mathematics and arithmetics, the origin of their problems is the crux of the matter. Their problems could be due to brain damages, poor teaching, the fact that we, simply, are good at different things, etc., etc.... However, in order for the phenomena dyscalculia and developmental dyscalculia to exist as concepts of their own, they have to arise from the most primeval learning disability in this line of categorization; and this would be difficulties in telling ’how many’ (cf. p. 13).

4 The Natural Number System: The Mathematical/Scientifical Perspective

In order to express these results in a fairly concise manner, we introduce the following notation. For a mathematical set \(\,S,\,\) we use the notation \(\,s\in {S}\,\) to express that \(\,s\,\) is an element of \(\,S\,.\) Moreover, the notation \(\,\forall {s}\in {S}\,\) will mean that every element \(\,s\,\) in \(\,S\,\) will satisfy certain condition(s). Finally, \(\,S'\subseteq {S}\,\) will mean that \(\,S'\,\) is a subset of \(\,S.\,\)

The fundament of the structure of the entire axis of the real numbers is the structure of its smallest subset with respect to the set inclusions

$$\begin{aligned} {\mathbb {N}}\subseteq {\mathbb {Z}}\subseteq {\mathbb {Q}}\subseteq {\mathbb {R}}, \end{aligned}$$

i.e., the natural numbers \(\,{\mathbb {N}}.\,\) The axiomatic representation of this number system is commonly known as Peano’s axioms, which are constituted by the following five postulates, in which the natural numbers \(\,{\mathbb {N}}\,\) is an unspecified set, where the existence of one natural number \(\,0\,\) (a first element) is postulated and a function \(\,s:{\mathbb {N}}\rightarrow {\mathbb {N}},\,\) which too is considered undefined. The axioms are the following.

$$\begin{aligned} \begin{array}{rl} \mathrm{(i)} &{} 0\in {\mathbb {N}};\\ \mathrm{(ii)} &{} s(n)\in {\mathbb {N}},\,\ \text {{ if}}\,\ n\in {\mathbb {N}};\\ \mathrm{(iii)} &{} \text { There is no element }\,n\in {\mathbb {N}},\, \text {such that }\,0=s(n);\\ \mathrm{(iv)} &{} s(m)=s(n)\,\ \Rightarrow \ \,m=n,\,\ \forall {m,n\in {\mathbb {N}}};\\ \mathrm{(v)} &{} \text {If }\,S\subseteq {\mathbb {N}},\, \,0\in {S}\, \text {and} \,s(n)\in {S},\, \,\forall {n}\in {S},\, \text {then} \,S={\mathbb {N}}.\, \end{array} \end{aligned}$$
(1)

Here, we thus particularly note that the natural number system \(\,{\mathbb {N}}\,\) in mathematical—scentifical—sense is constituted by an unspecified set, which has a first element—a ’zero’—a function \(\,s,\,\) which to each and every element of \(\,{\mathbb {N}}\,\) designates a successor \(\,s(n),\,\) which also, initially, is unspecified, and where two elements having the same successor by necessity must be the same. (v) is commonly referred to as the axiom of induction.

Note that in our usual representation of the natural number system \(\,{\mathbb {N}},\,\) \(\,s(n)=n+1.\,\)

To be able to understand the mathematical meaning of number, we introduce the following definition.

Definition 1

A function\(\,f:A\rightarrow {B}\,\) is constituted by three objects. Two non-empty sets \(\,A\,\) and \(\,B\,\) and a rule \(\,f,\,\) which to each element \(\,a\in {A}\,\) assigns a fully determined element \(\,f(a)=b\in {B}.\,\) These three objects determine the function \(\,f\,\) uniquely. The function \(\,f\,\) is said to be

  1. (i)

    one-to-one if for each element \(\,b\in {B},\,\) there is at most one element \(\,a\in {A},\,\) such that \(\,b=f(a)\,\);

  2. (ii)

    onto if for each element \(\,b\in {B},\,\) there is at least one element \(\,a\in {A},\,\) such that \(\,b=f(a)\,\);

  3. (iii)

    bijective if it is both onto and one-to-one, i.e., if for each element \(\,b\in {B},\,\) there is exactly one \(\,a\in {A},\,\) such that \(\,b=f(a)\).

Remark 2

In view of the above, we conclude that the mathematical meaning of a set consisting of a certain number of objects, e.g. 5 persons, is that there is a bijective function between the natural numbers 1, 2, 3, 4, 5 and the set in question. This means that we can pair the set of people together, one by one, with the numbers 1, 2, 3, 4, 5 and the rule f in Definition 1 is the order, in which this is done; the order in which we count the persons. If we replace the natural numbers 1, 2, 3, 4, 5 by another, initially unspecified, set, e.g. a set consisting of 5 rocks, this new set will serve equally well in keeping track of the number of persons in the crowd. One rock for each person and ’which’ rock given to each person is the rule f in Definition 1. This also yields another representation of the initially unspecified set in Peano’s axioms and a new function \(\,s\,\) yielding the successor of each element of the set, i.e., the order in which we distribute the rocks among the persons one by one. Hence, we have created a way of telling numbers, which is in no way connected to the written characters \(\,0,1,2,3,4,\ldots ,\,\) which we generally use as representation of the natural number system.

Though not of dire need for this article, we will also introduce the following definition of the binary relation of addition, which requires a little more than basic treatment of numbers (i.e., telling ‘how many’), and hence lies beyond the scope of dyscalculia.

Definition 2

For \(\,m,n\in {\mathbb {N}},\,\) we define addition recursively by

$$\begin{aligned} m+0= & {} m;\\ m+s(n)= & {} s(m+n),\quad {m,n\in {\mathbb {N}}}. \end{aligned}$$

Example 4

$$\begin{aligned} 2+3= & 2+s\left( 2\right) \\= & {} s\left( 2+2\right) = s\left( 2+s\left( 1\right) \right) = s\left( s\left( 2+1\right) \right) \\= & {} s\left( s\left( 2+s\left( 0\right) \right) \right) = s\left( s\left( s\left( 2+0\right) \right) \right) = s\left( s\left( s\left( 2\right) \right) \right) \\= & s\left( s\left( 3\right) \right) =s(4)=5. \end{aligned}$$

5 Analysis and Conclusion

Once again, the notations \(0,1,2,\ldots \) of the natural numbers rest completely on the nature of the characters, which we have made into standard in our development of the language as means of communication in writing. In view of Sect. 4, this is thus merely a representation, a presentation, of the unspecified set constituting the natural numbers \(\,{\mathbb {N}}\,\) according to the axiomatic representation of the same mathematical structure; and our way of counting, i.e., the order relation \(\,0,1,2,\ldots ,\,\) is merely a representation of the unspecified function (cf. Peano’s axioms (1) in Sect. 4) \(\,s:{\mathbb {N}}\rightarrow {\mathbb {N}},\,\) which to each element \(\,n\in {\mathbb {N}}\,\) assigns its successor \(\,s(n)\in {\mathbb {N}}.\,\) Thus, in order for dyscalculia to be able to exist in the sense that the experts of today claim it does (cf. i.a. Lundberg and Sterner 2009, pp. 17–21 and Sect. 3), the arisen difficulties in the fundamental abilility of counting, the ability of counting and telling numbers (i.e., ’how many’), i.e., the epitome of the natural number system \(\,{\mathbb {N}},\,\) must rest neither on the choice of representation of the unspecified set constituting the set of natural numbers \(\,{\mathbb {N}},\,\) nor on the choice of representation of the unspecified function \(\,s:{\mathbb {N}}\rightarrow {\mathbb {N}}\,\) (cf. (1) Sect. 4), which thus assigns to each element \(\,n\in {\mathbb {N}}\,\) its successor \(\,s(n)\in {\mathbb {N}},\,\) and hence, according to the axiom of induction (axiom \(\,(v)\,\) in (1)), constitutes the order between all the elements of \(\,{\mathbb {N}}.\,\) The persons with claimed dyscalculia must, by necessity, encounter with the same kind of difficulties independently of which ever representation we choose to adopt to concretize the mathematical structure of \(\,{\mathbb {N}}.\,\) In order to disprove the existence of dyscalculia as an own concept, it thus suffices to produce one representation of \(\,{\mathbb {N}},\,\) one counter example, where the persons with claimed dyscalculia will not experience any problems like with the representation given by the characters \(\,0,1,2,\ldots \,\). Our unambiguous conclusion will then be that the phenomenon of dyscalculia, its problems, are inherent in the nature of the representation of \(\,{\mathbb {N}}\,\) given by our developed charaters \(\,0,1,2,\ldots \,\). This phenomenen is commonly known as dyslexia. Note that people with dyscalculia, as well as dyslexia, are assumed to have an, in other parts, well-behaved intellect.

Theorem 1

Distinguishing between different natural numbers, i.e., between determining ‘how many’, is mathematically equivalent to orientating oneself in time, i.e., keeping track of the chronological order of events in a certain time sequence.

Proof

Let us first of all recall that, according to Remark 2, the distinguishing between different natural numbers, i.e., e.g. ‘Which pile of stones represents the largest natural number?’, is in no way mathematically in general connected to our representation of the natural number system \(\,{\mathbb {N}}\,\) by means of the written characters \(\,0,1,2,\ldots ,\,\) which we have developed as a means of communicating in writing. If we, in view of Peano’s axioms (cf. (1) of Sect. 4) consider each pile of stones as a subset of the intially unspecified set \(\,{\mathbb {N}},\,\) and if we line up the stones of each pile in parallel lines, from left to right, the stones together with the order, in which we line them up will represent a substructure of the natural number system \(\,{\mathbb {N}}.\,\) The pile of stones yielding the line with last stone ending up furthest to the right, thus represents the largest natural number.

Now, assume that the unspecified set (cf. (1) of Sect. 4) constituting the natural numbers \(\,{\mathbb {N}}\,\) is the set of all every day events—with the first element, the element \(\,0\,\) as our waking up in the morning—and the function \(\,s\,\) is the order in which we perform them during the day. This will yield another representation of the mathematical/scientifical structure of \(\,{\mathbb {N}}\,\) as described in (1), Sect. 4 This representation is not connected with a good ability to read and to write. The successor of 0 might be getting out of bed, which with our original representation would be denoted by 1. Hence, the telling of differences in ’size’—as described in Sect. 3—would consist in telling when, during the day, we carry out a certain chore, e.g. is the distance between getting out of bed and having morning coffee bigger than the distance between having morning coffee and having dinner, i.e., between which ordered pair of these events is the lapse of time the biggest, i.e., between which pair of events will the biggest number of other chores take place.

Thus, we conclude that distinguishing between different natural numbers is mathematically equivalent to orientating oneself in time, meaning the ability to chronologicallly keeping track of the different events in a certain time sequence. Thus, the theorem is completely proven. \(\square \)

Remark 3

Note that Theorem 1 is rigorously, mathematically proven down to an axiomatic level, which makes this result a result from a science—mathematics—where the causal structure and the ontological model are identical, while most results in general pedagogics research rest on a large amount of correlation.

Remark 4

In Sudha and Shalini (2014) they establish that brain screening reveals that when your right brain hemisphere is malfunctioning, the person suffers from lacking in understanding the concept of quantities as well as lacking in ability of learning sequencies in space (time) (cf. Theorem 1 above). Thus, these two abilities apparently emanate from the same brain area, which is substantiated by Theorem 1 above.

Remark 5

The test described in Example 1 would then look something like the following, where we are to tell between which of the chores having breakfast, going to work and going to bed at night we perform the most number of other chores, i.e., between which of the aforementioned chores is the lapse of time the biggest.

The congruent and the incongruent presentation of the problem in the test would then look as follows:

$$\begin{aligned} (\text {having breakfast},.............\text {going to work},.................................\text {going to bed}) \end{aligned}$$

and

$$\begin{aligned} (\text {having breakfast},.................................\text {going to work},............\text {going to bed}), \end{aligned}$$

respectively. The fact that, in the original test, as described in Example 1, where the test was conducted with digits/characters with the outcome that the test persons with claimed dyscalculia, and which are claimed not having developed a mental number axis, make less mistakes than the persons without claimed dyscalculia, would be a very strong indication that most people experiencing problems in counting and telling numbers are in fact having problems with the representation of the natural numbers \(\,{\mathbb {N}}\,\) (cf. Sect. 4), which we have obtained by developing our written language and the inherent characters we use, i.e., \(\,0,1,2,\ldots ,\,\) to represent the unspecified set in Peano’s axioms (1), Sect. 4. More precisely, persons with a mechanical, represented with characters and drummed into their heads, model of the number axis are very likely to make more mistakes in the test in Example 1 as a consequence of their mechanical knowledge leading to spontaneous answers without consideration and any deeper understanding of the subject. The person with claimed dyscalculia, however, who, for some reason has failed to develop a mental number axis is making less mistakes in this test... Let us establish that without a well developed, mechanically drummed into your head, mental number axis, you are constrained to think before yielding the answers to this test. This implies that the persons with claimed dyscalculia make less errors and have a higher frequency of correct answers. This would, without a too intrinsic, further analysis, bear testimony of a good understanding of—and a similarly good ability to treat—the mathematical structure (cf. Sect. 4) of the system of the natural numbers.

In the light of this, we come to the unequivocal conclusion that a person with dyscalculia—according to studies conducted within the area of expertise—would have enormous difficulties in keeping track of the order in which he/she is carrying out the everyday chores of his/her life. This means that these persons would have excessive problems of orienting themselves in time and space and in structuring their day. This, however, does occur with some people having brain damages or suffering from some kinds of mental diseases, but these diagnoses are not compatible with ’in other respects having a well-behaved intellect’. For further problems inherent in ’dyscalculia’, cf. Lundberg and Sterner (2009, pp. 16–19).

Remark 6

Moreover, Remark 2 gives the exact mathematical meaning of two sets having the same number of elements; they can be paired together two by two and no one will be left odd. If we reconnect to Example 3—Butterworth (2003) above—this test consists of, apart from the ability of telling ’how many’, yet another moment, another potentially difficult step—the ability of reading (and reading fast) the figure in the circle to the right; and after that combining this exercise in reading with telling ’how many’. That makes this test not a test of pure dyscalculia but more like a test of reading capabilities, and therefore a poor result on this test is inconclusive for telling that a person has difficulties in assimilating the essence of the scientific structure of mathematics, which would be dyscalculia. Instead, this test is more likely to measure the ability of handling the extra problems inherent in our elaborated representation of the system of natural numbers, which is a product of the characters we use for this purpose. That would be dyslexia.

Remark 7

Note that the recursive definition of addition from Definition 2 would work equally well with the set representing the natural numbers in Theorem 1, which would give us a completely analogous theory of the natural number system.

Remark 8

In the light of Theorem 1 and Remark 5 above, dyscalculia—as commonly studied and defined—can not be considered existent as a concept of its own. Dyscalculia in the mathematical sense would imply substantial restrictions in a persons concept of reality and intellect, which is not the assumed in the prevailing theories of dyscalculia. These theories are intimately connected with our representation of the system of the natural numbers, i.e., our expressing the natural numbers by means of the characters available, i.e., \(\,0,1,2,\ldots \). In the process reading and understanding a mathematical problem/elaborating/fomulating/writing/presenting a solution a person with dyscalculia would encounter with problems in the subsequence elaborating/fomulating a solution to a mathematical problem (cf. Sect. 1’When I have read the problem many times, then I understand and can solve the problem.’). Therefore, the phenomenon called ’dyscalculia’ does not mean problems in handling the mathematical structure (cf. Sect. 4)—the scientific quintessence—of the natural numbers, but in the way we communicate them, the way we represent the natural numbers, by means of the characters we use in communicating in writing. Thus, the phenomenon called ’dyscalculia’ is instead intimately connected with the extra set of difficulties inherent in our representation of the natural numbers by means of reading and writing them \(\,0,1,2,\ldots \). This would be dyslexia.

In Tambychika and Thamby (2010), the authors have performed a study reported from the students’ own experiences, where the researchers have measured how the students’ cognitive abilities operate. Firstly, they explain how problem solving is manifested and how the students experience difficulties. There are two aspects of problem solving—according to Tambychika and Thamby (2010) (cf. Sect. 1)—where the first is how the student understands/translates the problem linguistically (words) (cf. Sect. 2). The second problem solving issue is the non-linguistic aspect (cf. Sect. 3), which refers to numbers and graphs. Along with these two aspects, the problem solving difficulties can result in how the linguistic information is translated to mathematical terms (cf. Sect. 4)—numbers—and/or how the numbers should be solved/calculated. Students showed more difficulties in the first aspect, in which they transform the words into numbers and graphs. Again, this supports the fact that difficulties in mathematical problem solving is due only to dyslexia, and dyscalculia is a mere guise of dyslexia.

Bugden and Ansari (2014), argue that children and adults with dyscalculia suffer from different cognitive disabilities like for instance, both long- and short term memory struggles as well as difficulties in recognizing and translating the numbers to meaningful and useful information in the brain. In the case of dyslexia, the cognitive disabilities are similar as one’s ability to understand and transform letters into useful information, which fits with our memorized cognitive schemas. Bugden and Ansari (2014), have—like other researchers—shown how the non-diagnosed children’s brains work differently from the dyscalculia brain and stresses on the fact that the dyscalculia brain does not have any brain damage, it just works differently. Prominence is also given to the fact that, when dyscalculia is caused by cognitive disabilities, it is categorized as a secondary dyscalculia, which also explains that dyscalculia and dyslexia should be in one category, since the cognitive origin of the struggles affect reading all types of symbols and does not distinguish between letters or numbers.

Regarding developmental dyscalculia, almost all the experts in this area give prominence to the fact that an unambiguous definition of this phenomenon is nowhere to be found; the say that it is even dependent of in which age—and at which mathematical level—the different persons are. Szücs and Goswami (2013) list a number of different phenomena (Dyscalculia, Developmental Dyscalculia (DD), Arithmetic-related learning disabilities (AD), Arithmetical disability (ARITHD), Arithmetic learning disability (ALD), Mathematical Disability (MD), Mathematics Learning Disabilities (MLD), Mathematical Learning Difficulty (MLD)), which they state might or might not be equivalent to DD. However, the common treat of all experts opinions is that the primeval state of what they believe to be DD is the problem of telling ‘how many’, i.e., problems in handling the axiom of well-order, which is equivalent to axiom \(\,(ii)\,\) in Eq. (1) in Sect. 4. Thus, this is ultimately the core of what is called dyscalculia. Should this primeval state of the current learning disability give rise to more general problems in handling mathematics and arithmetics later on in life, the persons in question would also have problems in handling the axiom of well-order, i.e., telling ‘how many’, which thus is mathematically equivalent to orientate oneself in time and space. Moreover, in Centre for Neuroscience in Education (2020), it is explained that there are two predominant theories in this area. The first one believes dyscalculia to be the result of an impairment of the Approximate Number System, ANS, which equip humans with the capacity of quickly and instinctively understand, estimate and manipulate non-symbolic quantitites. This impairment would then, according to Theorem 1 and Remark 5, unequivocally affect the ability to orientate oneself in time and space, and this is not compatible with possessing an in other parts well-behaved intellect. The second theory states that dyscalculia impairment is not linked to the ANS per se, but that dyscalculics’ ability to automatically map symbols to their corresponding magnitude is disrupted. This would be encountering with problems with our chosen representation of the natural number system \(\,{\mathbb {N}}\,\) and is thus linked to the ability of reading and writing the numbers 0,1,2,... and decoding the information in the brain. This would be dyslexia.

Although there certainly and undoubtedly exist people with genuine problems in handling mathematics and arithmetics, the origin of their problems is the crux of the matter. Their problems could be due to brain damages, poor teaching, the fact that we, simply, are good at different things, etc., etc.... However, in order for the phenomena dyscalculia and developmental dyscalculia to exist as concepts of their own, they have to arise from the most primeval learning disability in this line of categorization; and this would be difficulties in telling ’how many’ (cf. Lundberg and Sterner 2009, pp. 20–24), i.e., problems with the axiom of well-order, which in turn—according to what we just have estasblished (cf. Theorem 1 and Remark 5) would be mathematically equivalent to problems in orientating themselves in time and space, which is not compatible with possessing an in other parts well-behaved intellect.

Altogether, we conclude that the quintessence of the phenomenon called ’dyscalculia’ is not inherent in the nature of mathematics/science (Sect. 4), but in our way of communicating mathematics/science, our representation of the system of the natural numbers by means of characters (cf. Remark 2, Sect. 4, Theorem 1 and Remarks 5–7, Sect. 5). Altogether, ’dyscalculia’—as today classified and defined—can not be considered existent but is merely just another one of the sad guises of dyslexia.

These conclusions are also substantiated by the experimental results found in Peters et al. (2018), where children with one—or both—of the two learning disabilities dyscalculia and dyslexia evince the same type of brain activities, located to the same areas of the brain, when doing arithmetics.