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Is there a way of more advanced minds start to count with complex integers, instead of naturals? This way such a society would never think of $2$ as prime since $2=(1+i)(1-i)$. Notice that more advanced societies may have some shortcomings in thinking and art. For instance, try of language without knowing the existence of writing, and how many languages flourished since many people in the middle ages didn't know to read.

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    $\begingroup$ "complex integer" is a contradiction in terms, mathematically speaking. $\endgroup$
    – L.Dutch
    Sep 13 '20 at 13:25
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    $\begingroup$ They are actually called Gaussian integers. I guess it would be natural for a society who would think bidimensionally or multidimensionally. $\endgroup$
    – checkmath
    Sep 13 '20 at 13:29
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    $\begingroup$ What would be a (1+i) apple? $\endgroup$
    – L.Dutch
    Sep 13 '20 at 13:34
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    $\begingroup$ I can't imagine why anyone would do this. We've known about N-dimensional vectors for some time (a complex number is homologous to a 2-vector), and while they have uses (computer graphics, anyone?), scalars are still appropriate in many, many instances. I doubt anyone here will be able to come up with a brilliant and indispensable use given that all of humanity has failed to do so in centuries. $\endgroup$
    – Matthew
    Sep 13 '20 at 14:57
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    $\begingroup$ You cannot "count with complex integers". Complex numbers do not have an order relationship. That is, it is meaningless to ask whether a complex number is greater than another. And about that example with apples: how do you represent two apples collected two days ago? And why is it different from 2(1 + 2i) apples? $\endgroup$
    – AlexP
    Sep 14 '20 at 7:09

11 Answers 11

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Complex numbers provide answers to a different set of questions than those that arise with basic counting, so it is indeed unlikely that any civilisation would invent those before the natural numbers. The human history of mathematics does provide ample examples of things being discovered in what to modern eyes looks like the totally wrong order — real numbers were known from Greek Antiquity (with Eudoxus providing a surprisingly rigorous characterisation) whereas the decimal fractions taught in school today is a 16th century invention, but then again already the Babylonians essentially did floating-point numbers (except they only did it in base 60, so for millennia everybody else only did so too) — but I'd recommend reading up a lot on both the history and the mathematics before trying to craft your own timeline, just to get a feeling for what might be reasonable and what is not. That's probably not the answer you're looking for.

If instead we take your question to be one of "how could one get complex numbers more integrated into basic arithmetical practices?" then one approach could be to use a complex-base number system, for example the base $-1+i$ binary system where $2 = (-1+i)^3 + (-1+i)^2$ is written 1100; this provides a unique encoding for precisely the Gaussian integers. Why would a civilisation employ such a curious system? Well, in the early days of computer languages it wasn't obvious that you might have use for several different kinds of numbers in computing; many languages just had one "number" datatype, which in languages designed for scientific applications was "real" numbers (under the hood they might be floating-point or fixed-point, sometimes depending on the compiler). Real numbers were what human engineers used, but if there had been a significant need for complex arithmetic instead (say, if the early computing history of your aliens was dominated by quantum mechanical problems rather than ballistics), then it's not a big stretch to offer complex numbers as the Standard Number Concept in your system — even in our civilisation Computer Algebra Systems often default to "every variable is complex, until you specify otherwise" which sometimes leads to results surprising users who expected real variables. Since memory management was tricky in early computing, it is also not unreasonable that they might want to make a complex number that is not built up from two separate pieces, and then base $-1+i$ is a neat solution.

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Tl;DR: I would argue the complex numbers are unlikely to be used, because I can't see a simple (or even consistent) use of their particular multiplicative structure.

Let's consider one of your examples in the comments, the case where the complex number $z_1=3+70i$ represents a family with 3 family members and a total age of 70 years. If we have a second family of 2 people with a total age of 50 years, it can be represented as $z_2=2+50i$. Adding these together yields $z_3$, a collection of $3+2$ people with a total age of $70+50$ years, represented as $z_3=5+120i$.

But wait a second. We haven't invoked any special properties of complex numbers; we've simply done pairwise addition. Here, the complex numbers under addition are equivalent ("isomorphic", in formal terminology) to the two-dimensional real plane, $\mathbb{R}\times\mathbb{R}$, under addition. Indeed, the same should hold for any collection of things that can be viewed as a vector space with two dimensions. In other words, by using complex numbers, we obtain no benefits when we add things together.

The question, then, is why we should choose the multiplicative structure of complex numbers - and from here on out, I'll represent $z=a+ib$ as $(a,b)$, to drive home the point.$^{\dagger}$ Let's say we again have $z_1$ and $z_2$, as defined above - two families of people. Now let's multiply them. We then get $$z_1\cdot z_2=(a_1a_2-b_1b_2, a_2b_1+a_1b_2)$$ Here's the issue: The quantity $a_1a_2$ has units of $(\text{people})^2$ (or, arguably, no units at all). On the other hand, the quantity $b_1b_2$ has units of $(\text{years})^2$ - and dimensional analysis says we can't add together two quantities with different units. In other words, complex multiplication, in the way you've defined it, doesn't make logical sense.

In fact, the same holds for many different arbitrary types of multiplication you can define on a given vector space - where by "multiplication" I mean a particular type of binary operation that at some point involves multiplying (in the one-dimensional, real number sense) components of two different pairs together. While it's conceivable that a society may find it useful to consider two properties of an object when discussing it, I'm not sure that complex number multiplication is the correct definition of multiplication you're looking for. For instance, why not define multiplication by $$(a_1, b_1)\cdot (a_2, b_2)=\left(\frac{a_1a_2}{a_1+a_2},\frac{b_1b_2}{b_1+b_2}\right)$$ which does, of course, preserve units correctly? (Although in this system, there is no identity element.)

There are other, related operations that make sense, though; for example, consider scalar multiplication. If we multiple $z_1$ by the real scalar 7, we get a quantity that does have a concrete physical interpretation: namely, a family of 21 people (yikes!) with a combined age of 490 years. On the other hand, in this case, there's no advantage to using complex numbers over simply $\mathbb{R}\times\mathbb{R}$.


$^{\dagger}$Pun intended.

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  • $\begingroup$ There might be some need for intuitive complex thought if the creatures in question view the world through some transform, but that’s a very different situation from the OP. $\endgroup$
    – Joe Bloggs
    Sep 14 '20 at 11:10
  • $\begingroup$ If someone is unsure what the problem is with adding different units: consider that you can measure distance in either centimeters or kilometers, and you can measure time in either seconds or hours. We know how to convert between cm and km, and between sec and hr. Because these conversions are ratios, we also know exactly how to convert from $\dfrac {cm}{hr^2}$ to $\dfrac {km}{sec^2}$. But how would you convert from $cm + hr$ to $km + sec$? You would need to multiply the cm portion by a different ratio than the hr portion, but when what you know is the sum, you have no idea how much each was. $\endgroup$ Sep 15 '20 at 2:21
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To understand complex numbers, you need to understand positive real numbers first.

You can define a number on the complex plane in 2 main ways:

  • Coordinates: 4 + 2i. 4 steps in the + real direction, 2 steps in the i direction.
  • Polar: 3e^(ipi1.5). 3 steps in the direction of 1.5 radians (mostly i direction).

Note that both of these methods require an underlying concept of real numbers. If i wanted to encode a complex number which was (2 + i) steps in real direction and then (3 - i) steps in the i direction, I'd have (2 + i) + (3 - i)i. Which is 2 + i + 3i + 1, or 3 + 4i. To comprehend 3 + 4i, you need to be able to comprehend, 3, 4, and i.

Complex numbers don't get a real use case until your society has advanced a long way.

We didn't start using them until the 16th century (at best), we had discovered algebra, some trig, polynomials, and pi to 16 places.

Most people don't use complex numbers after high school. I've used them a few times in electrical engineering, in signal analysis, and writing lossy compression for voxels of 3D mining survey data, but, I'm not a typical person.

Complex numbers are less efficient in digital systems than real integers.

If you're developing a first generation computer, every bit counts. The large subset of operations which can be done with i=0 would be a substantial optimisation for your early computer code.

Counting in complex integers is ambiguous and requires moving in +, -, +i, and -i directions:

Counting in real numbers is easily defined. I_next = I_current + 1.

Counting in complex numbers has multiple directions. You're going to need to count in all 4 directions (you can't just count in only +i and +1, as counting must continue infinitely, and you can't count to 0i + infinity and then wrap around to 1i + 0).

I can only see four "practical" methods of counting in complex space:

  • What I'm going to call "Snake" counting, where the previous value never changes by more than 1 (or i) from the previous value (It's like playing the game "Snake"):
    enter image description here
    (Follow the alphabet round the 2D space to count, eg 0, 1, 1 + i, i, 2i, 1 + 2i, etc)

  • 1 quadrant sorted by absolute value
    enter image description here

  • That again but flipped along the diagonal, so 0, 1, i, etc. instead.

  • 4 quadrants, sorted by absolute value, which is also continous
    enter image description here

Complex wont happen, but tuples might be nice.

Having a counting system which changes between those 4 headings in the complex plane would be a headache, and I can't see the gain. Especially as when the number system is being worked out, the population are counting loafs of bread or number of coins.

You mentioned in the comments that you thought of examples like "1 + 2i apples, could mean 1 apple collected 2 days ago". This isn't very handy to think of as a complex number, because operations don't apply to it. (1 + 2i) + (2 + 3i) is 3 apples that are 5 days old, which is wrong, if anything you have 3 apples averaging 2.66 days old. The + operator became an average operation in this context, whereas in other complex cases this wouldn't work.

However if I were to request a maths upgrade to my brain, the ability to think in tuples and vectors like you mentioned in the comment would be very handy:

For example: I have \$(100, 20, 50, -80) dollars in the bank. That means I have 100 in my day to day, 20 available on my credit card, 50 coming in tonight, and 80 going out tonight. If I pay these 3 bills now it'll cost me \$(40, 10, 0,-30), which is 40 on my day to day, 10 on my credit card, and 30 that was going to come out tonight to pay one of them no longer will. This leaves me with \$(60,10,50,50). It'd be pretty nice if I could work that out without having to subtract each one individually.

This is a bit of a contrived example, but thinking and processing in parallel like that could be pretty useful, and it may improve the understanding of complex numbers, however it wont replace the need to define both tuples and complex numbers on top of the fundamental positive real numbers.

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  • $\begingroup$ Excellent answer. Complex numbers just don't make sense for everyday usage (except maybe if time were multidimensional, then they might be useful for timekeeping), but better mental ability to work with vectors (or better yet matrices) would be a huge boon for a large number of reasons (speaking here as someone who actually has an above average talent for vector and matrix math, it's been very helpful in a number of cases that many people would not expect). $\endgroup$ Sep 14 '20 at 1:50
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Leopold Kronecker is quoted as saying "God made the integers, all else is the work of man." Your aliens might replace that with "God made Gaussian integers," but they will still study ordinary integers.

When you get down to it, mathematics is the science of evaluating models that are based based on axioms and drawings conclusions whichare more (or less) relevant to real life. If you learn mathematics rather than primary school arithmetic, you usually start with extremely simple axioms and models and then progress to more complicated ones. And when it comes to applied mathematics, you should always try if the simplest models give interesting results before you try to fit a more complicated model to the facts and problems. That applies in calculus, geometry, set theory, graph theory, statistics, you name it.

The concept of a prime matters when one does prime factorization, for instance, and that in turn is useful for cryptography. Not considering 2 a prime number does not change the fact that such numbers are useful -- you would have to invent a new name for the property of integers being composed that way. Or take group theory.

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  • $\begingroup$ I'm pretty sure that ℕ ("natürlichen Zahlen") and ℤ ("ganzen Zahlen" = "whole numbers") are different sets even in German. :) $\endgroup$
    – AlexP
    Sep 14 '20 at 7:17
  • $\begingroup$ @AlexP, oops, fixed. $\endgroup$
    – o.m.
    Sep 14 '20 at 8:13
  • $\begingroup$ I wouldn't worry so much about cryptography, which rarely cares about details like "Gaussian integer" vs "integer" - in fact, the amount to which it does not matter is often surprising. There are some nuances here, but, on the whole, cryptography cares that you're picking prime elements in whatever context you're working in - as long as your notion of "prime" matches up to your notion of "integer", you're probably okay. $\endgroup$ Sep 14 '20 at 23:09
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I doubt it. The trouble is that the natural numbers - that is $1,\,2,\,3,\,4,\,5,\ldots$ and sometimes $0$ - have so many special properties. In the scope of human mathematics, there are many fields that do not care about real numbers or rational numbers or even negative numbers - but the natural numbers seem almost inescapable just because they are so many things.

To start out with:

  • Natural numbers are cardinal numbers. A natural number answers questions like "how many coins do I have?" or "how many people are in this family?". Plus, the operations of natural numbers come with this - I can naturally ask "how many outfits can I get by combining advanced-society-pants with an advanced-society-shirt?" which corresponds to a question of pairs about sets ("cartesian products") or a question of multiplication of natural numbers. Similarly, "if I steal all of Hannah's advanced-society-money and combine it with my own, how many coins would I have?" relates addition and the notion of combining two sets ("disjoint union").

  • Natural numbers are ordinal numbers. Natural numbers are used to rank things - if I have a set of preferences, I can use them to refer to my first preference (the best option) or second (the best option other than my first choice) and so on.

  • Natural numbers can index repetition. If I want to specify "repeat something a certain number of times" I can specify this using a natural number. This is even ingrained into notation of math - if I want to add something together a bunch of times, I might write $5x$ even if multiplication by a natural number is not strictly allowed and I really mean $x+x+x+x+x$. Similarly for $x^5$ or even with function iteration $f^5(x)$. In higher mathematical terms, this is related to the fact that you get the natural numbers if you say* "There is some natural number zero and each natural number has a successor" - as is often used to axiomatize natural numbers.

  • Natural numbers are the (second) simplest example of arithmetic. In more abstract sense than the previous ones: natural numbers have nice abstract properties. You want a number system where you can add any two numbers and you want associativity ($a+(b+c)=(a+b)+c$)? Well, you can either go with the advanced-society-nihilism of "there are no numbers" or the slightly more complex "well, there's a number $1$... oh, and there'd better be $1+1$ and $1+1+1$ and so on too I guess." Same thing if you ask for associativity and commutativity ($a+b=b+a$). You can add an additive identity ($0$) if you like - same basic structure, though. If you want multiplication with distributivity ($a\times (b+c) = a\times b + a\times c$), same thing. You can even get your multiplicative identity (and by this point, you're asking that some specific numbers exist - meaning that "there are no numbers" is no longer a simpler system). This reason might not hit common people as much as the previous three, but it'd be hard to develop algebra without someone noticing these things.

I think it's essentially unavoidable that natural numbers will come up as the most fundamental system of numbers - and in any case, if they don't, their replacement would have to be a far more radical rethinking of mathematics and logic than "they use complex numbers instead." It's certainly possible to, for instance, never use cardinal numbers and to instead exclusively talk about sets themselves - and this is, in fact, very often a good idea both for ease of reasoning and ease of interpretation. Similarly, ordinals can be talked about without numbers - it certainly suffices to just say "here's some set, and here's some way to tell which elements are better than which other ones" - although this seems to get a btg more troubled (being that "3rd best" kind of means "the best element among those other than the absolute best element and the other element which is the best among those other than the absolute best") - but, regardless of specifics, the point here is that mathematics is very interconnected; changing everything at once might be plausible, but changing one detail is not.

That said, if you're willing to admit that some bits are the same, there's lots you might imagine gets done differently - even among human mathematicians, there's a lot of different views on mathematics; to name a few, there are philosophies such as constructivism which, essentially, rely on finite structures with clear rules to build up logic. There's a lot of a somewhat scientific or engineering perspective where computation is prized over other things (e.g. writing out problems via linear algebra or numerically solving differential equations - where some sequence of calculations is implied). There's some very abstract perspectives where one talks about what objects "do" more than other things (e.g. when one speaks of objects like transformations of the plane or symmetries without great detail on coordinates or stuff like that - category theory is a particularly broad example of this).

Exactly where you go after the natural numbers is going to depend a lot of philosophy - and there are certainly directions that would quickly lead to complex numbers if done motivated by academic interest and, if periodicity is a frequent part of life, one could imagine certain fields of mathematics that rely upon complex numbers (e.g. Fourier analysis) being so darn important that everyone knows about them (maybe they all play advanced-society-theremin and want to impress each other with awesome signal processing?). There's perhaps some other options where, for instance, cardinal numbers become irrelevant because most sets they interact with are somehow weighted by complex numbers - maybe if they're all trying to program quantum computers, they're suddenly going to feel that complex numbers are rather important, the same way that far more people know about binary in this world not because it's some fundamental mathematical idea, but just because it's our machines everywhere.

(*With some subtleties here - the properties mentioned relate to the natural numbers being "freely generated" by a single element zero and a successor function)

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No, there is not, for a very simple reason: the Gaussian integers do not have a unique total ordering.

Thus, there is no way to count with them.

Thus, "advanced minds" could not start to count with them.

Advanced minds might naturally comprehend the structure of the Gaussian integers as easily as we can be taught to work with normal integers, but they would still count with the counting numbers--non-negative real integers.

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The natural numbers are ideal for counting because one can define them to be the "simplest" set that has a unique total ordering. 2 >= 1 because 2 is defined as the smallest natural number greater than or equal to 1. The act of counting is just the definition of an injection of the natural numbers to an arbitrary set (such as the set of apples in your basket). If you have 7 apples, it is because you can assign each apples a different natural number from the subset {1, 3, 2, 6, 4, 7, 5}.

The complex numbers do not have a unique total ordering. If you represent them in a table

    0    1     2     3   ...

0   0   0+i   0+2i  0+3i ...
1   1   1+i   1+2i  1+3i ...
2   2   2+i   2+2i  2+3i ...
3   3   3+i   3+3i  3+3i ...
.
.
.

Then there are at least four trivial total orderings:

  1. 0, i, 1, 2i, 1+i, 2, 3i, 1+2i, 2+i, 3, ...
  2. 0, 1, i, 2, 1+i, 2i, 3, 2+i, 1+2i, 3i, ...
  3. 0, i, 1, 2, 1+i, 2i, 3i, 1+2i, 2+i, 3, ...
  4. 0, 1, i, 2i, 1+i, 2, 3, 2+i, 1+2i, 3i, ...

All four count along the diagonals of the table, either

  1. Always from the left up to the right
  2. Always from the right down to the left
  3. Alternating right and left
  4. Alternating left and right

An advanced civilization might have a use for all four orderings, or for even more complicated orderings as well.

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You are setting the abstraction bar quite high in your title, but then go on to lower it by providing a concrete example. Bidimensional thinking may not be what you imagine it to be.

Complex numbers do not represent bidimensional thinking, they represent some kind of spatial-algebraic formalization... the key is in thinking (which is, generally, not understood as a linear process, expressible through additions, multiplications etc.).

Bidimensional thinking is already "a thing" in our very own real world, in at least one sense. Take, for example, synesthesia. While certain important, utterly complicated, sensory neural pathways (as in "areas of the brain") are carefully isolated from others, our brains occasionally screw up and mix the pathways somehow. So, for example, there are people who perceive colors in numbers/symbols. When trying to utilize their "thinking" capacity to perform any analysis on those things, inadvertently, they are bound to exhibit some sort of "bidimensional thinking".

Let me give you a(n overly) simplified example (demonstrating a very real scientific question, though)... If a person perceives 4 as red and 5 as blue, by which color are they going to perceive the result of the operation 4 + 5? Let me just warn you by saying that this is quite the can of worms and no easy explanation exists, let alone a model of function that could deterministically describe how this kind of multidimensional thinking works.

Now, let me give you another example, one that you can probably "feel" yourself to understand the concept a bit better. Could you add words in the way that you can add numbers? Well, probably not... well, that is unless you have a certain dimension to express the result. Consider taste to be the dimension and try calculating the result of the operation:

salt + pepper

Can you "feel" that? Congratulations! You have just thought "bidimensionally". Now, there are people that don't even need this downplaying (i.e. limiting to the words that represent familiar ingredients associated with taste). They can taste pretty much any kind of word.

See, the thing is that thinking in complex numbers is not really bidimensional thinking, it is twice-unidimensional thinking, with some special adaptations/formalizations. True bidimensional thinking (the kind a world-building exercise could benefit from) would entail dimensions that operate pretty much differently to each other.

The hard benefit from managing to manipulate this "power" would be an unimaginable increase in problem-solving capacity and expressive power. Suddenly, (4 + salty) will be an element and combining this with (160 + mildly sweet) might be the solution to some specific version of a fair cake-cutting problem. Who knows...

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For quotidian purposes your system fails Ockham's Razor and would not pass into widespread use.

Instead I suggest matching rather than counting. It is sufficiently simple that children invent it independently and is often still used by adults when counting coins: count one stack then make the other stacks the same height. Then line them up in groups of two by five (dominantly matching instead of counting) because you can see whether you have ten instead of needing to count them.

One significant advantage of matching over counting occurs when dealing with infinities. Ordinal infinities cannot be compared, but this is an artefact of ordinality rather than infinity. For details research transfinite numbers.

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I am going to go halfway against the flow here. No, they are not going to develop complex numbers first, but it is not unreasonable that they could have developed them much earliar than we did. I think there is only one truly novel idea missing from the ancient Greek mathematicians that kept them from discovering complex numbers.

Mathematics arose out of a need for "inventory control" - that is, for keeping track of your stuff. That basic need is not going to be different for any creature we can make sense of. That your "advanced aliens" would be so stunted in their intellect as to not be able to notice 1-dimensional things, unless us who think easily in 1, 2, or 3 dimensions, and can conceptually deal with any number of them is ... difficult to understand.

But they could have encountered complex numbers far earlier in their mathematical development - though it seems unlikely they would do so before primes. We invented/discovered complex numbers in order to solve cubic equations. But supplying polynomial roots is far from the only use they have. Complex numbers revolutionized the subject of analytic geometry in the plane. Complex addition is just translation. Complex multiplication is a combination of rescaling and rotation. Translations and rotations are isometries. Rescaling is a similarity. The ancient Greeks were well-familiar with these.

If somehow the Greeks would have gotten the idea of that actions were not just processes to be applied to planar figures, but were objects of study in their own right - they might have discovered complex numbers themselves, developing them as an "arithmetic of planar transformations." Studying how rotations interact with each other, and combine to form other transformations might have led them to the concept of multiplications and all the various rules.

While it is hard to see how the Greeks might have come up with the concept of transformations as objects, a concept that comes to us only by way of centuries of additional mathematical development, it is not hard to imagine that a sufficiently different culture might find the idea far more quicker than we.

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slight frame challenge

"Is there a way of more advanced minds start to count with complex integers, instead of naturals?"
Why not "Is there a way of more advanced minds start to count with REAL numbers, instead of naturals?".
We use integers because that is how our mind works when processing the input from our senses. We see two apples, three dogs... We abstract, simplify and categorize. In that way our mind manages to deal with reality in real time.
But an alien mind might perceive reality differently and so quantify it with real numbers.
Human: "there are two apples on the table"
Alien: "there are 0.65 kg of apple material on a wooden support that is 142.5x80x80 cm in size"

It's like in that movie

Teacher: Who can tell me how many of these numbers are divisible by two?
Teacher: Anybody?
Teacher: Fred!
Fred: Hm?
Teacher: I know that you can tell me how many of these numbers are divisible by two.
Fred: Um... All of 'em.

When you change the way you look at things, the things you look at change
Wayne Dyer

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