From my philosophy, I would say "2" and your imaginary number "i" are exactly same existence relative to Math realm. Back to our common physical world since there're no two EXACTLY same leaves, so how even you can be sure to say "2" is absolutely real, and "i" is not?

Once you enter Argand plane, u can easily feel "i" as rotation by a right angle as one of its metaphysical explanations. "2" can also be explained as doubling an existing action as a function, not just a static common objects. So although in physical world "2" does not really exist, I think every sensible person will agree there exists a necessary mathematical concept called "2" after seeing two seemingly identical things. The only critical insight in this simple-yet-seemingly-philosophically-deep problem is that all human concepts/definitions are relative, meaning the concept "exists" itself is relative to a certain layer of our mind... "2" is certainly more abstract than "pain" which our body can feel impressively while a number, not so much due to our biological design. While certain math genius may feel "2" more impressively than "pain", and that's why this person can surely outperform you in solving Math number theory problems in the long run as a persistent game, unless somehow later you acquired more intuition after huge effort and struggle...The most hard part is when most people say something "exists" they implicitly assume there's an absolute background reference frame in which there's an objective yes or no binary definitive statement can be made. And most of them will be satisfied spending all their life arguing about this type of "wonderous" existence and that kind of "pitiful" nonsense, essentially much like machine, they'll keep their focus on these outward worldly existences to try to "prove" or "disprove" from their experiences as a vanity show to others. While for those rare illuminated and awaken people, inward retrospect and self-reflection is much more important than those outside existence or not. They fully understand when indulging in outward existence discussion the endgame is just to find a place for inward sentiment to attach, and they never lack any wealth to find such a place to know themselves much deeper and clearer. Neither outward realism nor nominalism is all the truth, they're just a small starting part of it to stimulate and invoke your inward "intellect memory sea", which in some religious factions it's called “the Eight Consciousnesses”. it's beyond common outward/inward perceptions, ego and comprehensions, similar to western world's subconsciousness notion. So the whole truth remains extremely hidden and mystically elusive in this vast sea of pictorial metaphors, no two persons will share exactly same images to the same clarity degree, not even nowadays super AI/GPT3 can sort out completely. Thus to understand and progress oneself accordingly is the ultimate goal and the only important truth for oneself...

By the way, most people will regard math as absolute truth, such as number 2 is real and really exists in some Platonic spiritual world apart from this imperfect material world (sounds like dualism here). But my view is contrary, number 2 (or any abstract universal concepts) resides in the same "metaphoric" realm of human mind, just happen to be located in the relatively most clear-countably-verifiable-universal layer. Again, literally written as an Arabic symbol "2", its semantic meaning can be limitless, it may be forming 2 dollars (object) in your mind, while doubling an existing action (function) in my mind, etc. When we talk about "2" in pure math context, the aforementioned concrete differences disappear, and it suddenly becomes a more clear-but-abstract notion which is still a metaphor but with much higher clarity compared to previous physical images. So in the physical context, it makes sense for nominalists to claim "2" does not exist at all (since relative to this context there're no two exactly identical leaves in this measurable physical layer), however, relative to the more abstract math layer, it also makes sense for realists to claim "2" exists in this Platonic realm... I don't opine separating the noumena from the phenomena as a serious business, its useful for some purposes, but all these concepts and separations are still man-made (fake) analogies consciously engineered to explain to a naive but confused child who is actively seeking an authoritative answer from the grown-ups. A child usually will be overwhelmed if provided too many alternative different explanations...

This world perceived by human mind is nothing but metaphors, that's why we can have several different models/theories about the same phenomena, such as the Newtonian Force Laws, Lagrangian/Hamiltonian Path Integral Minimum Action Principle, and the later Maxwell/Einstein Local Field Theory in classical physics and then applied further into QM, so far all these above 3 distinct models (metaphors) are not proved wrong while very useful and taught in every physics department around the globe. In the meantime, because our mind is constantly forming-destroying-reforming numerous metaphors as free will, most of these created images/processes/analogies are in more or less confused state. For example, if you've never been visiting a place and people around you are talking extensively about it, still in your mind you'll form some vague images from what you heard. Most of these misconceptions are like "avidya" in eastern Buddhism's nothing-but-metaphoric teachings, huge huge and thick darkness in the form of ignorant confusions is covering human mind like the "five mountains" and thus all its derived senses...

One of my favorite thinkers about the relationship of mathematics to other sciences is Charles Sanders Peirce. In Elucidations podcast Episode 81 the host speaks with a Peirce scholar, Cathy Legg, Lecturer in Philosophy at the University of Waikato, Hamilton, New Zealand.

Here is Kathy explaining the relevant passage from that discussion (starting ~24:50)

“Perice used the term architectonic which is a term in Kant. And its just like architecture is building a house and planning the overall structure. Architectonic is the same thing for knowledge.

And so Perice arrange what he called a hierarchy of the the sciences. And the basic structure was meant to be that they were sort of arranged one on top of the other. In a tower. And each science. And this is science in a very broad sense. Just meaning knowledge. Each science gets principles from the science above it in the hierarchy. And it takes data from the science below it in the hierarchy. So basically knowledge is inserting. Concepts are being inherited down this tower. So the basic structure of this tower is right on top is mathematics.

So this is a purely formal science. And mathematics. Perice had a hypothetical interpretation of mathematics. So mathematics doesn’t talk about what’s actual at all. Mathematics makes no positive claims. Mathematics just tells you if, you make this hypothesis, then this must follow. So mathematics is the science that draws necessary conclusions.

Directly after that Peirce put a science of phenomenology, which is the science, or the process of just sort of opening your eyes and looking around you and describing what you see. [...] Next up we get the three normative sciences. So these are aesthetics, then ethics, then logic. [...] And then from logic we get to metaphysics. And from metaphysics we get to physics and then we start with the natural sciences."

For Peirce, writing in the late 19th century, "mathematics makes no positive claims" to capital-T truth. Reality, or the natural sciences, only occurs later in his "hierarchy". And why not? Wasn't it Newton who thought to question his perception of reality by sticking a bodkin under his eyeball?

You are starting from a correspondence theory of "Truth," in which truth is defined as a statement corresponding with a state of affairs in the real world (taking for granted that a non-problematic real world exists, and that it's one most familiar to us). However, i by itself is not a statement, and "1 + 1 = 2" is not a statement about the world, it is a statement within the system of mathematics.

To make it a statement about the world, you'd have to have to map it to the world. "1 apple + 1 apple = 2 apples" is a example of a hybrid statement, combining mathematics and real world objects, and it seemingly meets your definition of true.

Certainly, statements like "2 + 3i apples - 1 + i apples = 1 + 2i apples" don't appear to make much sense. But it has been demonstrated that imaginary numbers do correspond with things we encounter in the real world, such as electricity. And in fact, many of the most arcane and grotesque corners of mathematics have proven, in the fullness of time, to describe one or another aspect of ordinary existence. With that said, "Mathematics is truth" or "Mathematics is not truth" are not sentences that even make sense under the definition of truth that you have provided. At most, you could say that some hybrid mathematical/real-world sentences are true, in as much as they correspond to some actual state of affairs.

Negative numbers could describe the properties of antimatter. Antimatter destroys or nullifies mater. Mathematics and physics are languages that together can describe the physical phenomena of matter. But no language is ever truth. The direct experience of reality far greater than any language. Language is only a map of reality.

tl;dr- Having i apples is just gibberish, like having ZOOOOM! apples is gibberish. That the gibberish includes math-related language isn't really relevant.

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Having i apples is gibberish.

You can't have i apples for the same reason you can't have AHHHH-IS-THAT-AN-EEL-ON-YOUR-FACE?!?! apples: because the statement doesn't mean anything. It's literally just gibberish.

I mean, sure, we try to coerce statements into something that makes sense. In the case of

I have [x] apples.

, we can try to coerce various unconventional expressions of "x" into something that makes sense. Examples:

1. Zero apples can be understood as not having any apples.

2. Negative apples might imply owing someone else an apple, or perhaps having an apple composed of anti-matter, depending on context.

3. A fraction of apples might imply having part of an apple, or perhaps an apple that's some fraction of a standard-apple's size, depending on context.

4. A mixed number of apples might imply having a certain number of whole apples, and then part of an apple, or perhaps some summation of partial apples, depending on context.

   • For example, if someone says that they have "1.5 apples", then they're clearly not talking about just whole apples.. so maybe they have 3 half-apples?

And, heck, having i apples could make sense in some esoteric contexts!

5. A non-real complex number of apples might imply a quantum superposition of apples, or maybe apples in some cyclic process (e.g., as with imaginary electricity).

Still, language is about communication, so just because having i apples could make sense in some contrived contexts doesn't really change much in a general context.

Typically, a reasonable reader should look at a statement like

I have i apples.

and conclude that it's gibberish rather than trying to reach to coerce it into something meaningful, unless they have some basis for justifying whatever meaning they might attach to it.

In short, it's essentially just gibberish. Sure it contains something that sounds math-y, but adding a mathematical expression in a nonsensical way doesn't turn something into math.

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Prior work

Here's something from Wikipedia about prior publications on this topic:

Colorless green ideas sleep furiously is a sentence composed by Noam Chomsky in his 1957 book Syntactic Structures as an example of a sentence that is grammatically correct, but semantically nonsensical. The sentence was originally used in his 1955 thesis The Logical Structure of Linguistic Theory and in his 1956 paper "Three Models for the Description of Language". Although the sentence is grammatically correct, no obvious understandable meaning can be derived from it, and thus it demonstrates the distinction between syntax and semantics. As an example of a category mistake, it was used to show the inadequacy of certain probabilistic models of grammar, and the need for more structured models.

–"Colorless green ideas sleep furiously", Wikipedia [references and links omitted]

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Analogy: Weak numeric typing in computer programming

In computer programming, we might say that a value for a time's minute-count should be an int, but this doesn't mean that any int is a valid value for a time's minute-count.

If we were being precise about everything, we shouldn't store a number-of-minutes as an int, but rather as a NumberOfMinutes. Most folks don't bother to be that explicit in their type-definitions, though; it's commonly accepted to use a primitive-numeric-data-type with data-validation-checks to hack-in strong-typing functionality without actually bothering with strong-typing.

Likewise, having i apples can be seen as having [scalar] apples, where we're disregarding that it should really be more of [number-of-apples] apples and trusting the surrounding logic to make up for the deficiency in strong-typing precision.

From this perspective, we can describe a statement about i apples as having a type-error.

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Tangential: Oxymoronic wise expressions

While I'm weirdly having trouble Google'ing this, I've seen a lot of old-fashion "wise" expressions like:

1. Only the blind can truly see.

2. Only the deaf can truly hear.

3. Only the penniless can truly be wealthy.

4. Only can fools know true wisdom.

5. Only can failures truly succeed.

Expressions like this seem to be regarded as wise despite being kinda stupid.

I think they work because, if you assume that they're meaningful, then you have to think hard to try to figure out some coherent meaning to assign to them. This can invoke an extensive brainstorm, as the mind tries to come up with the least absurd interpretation.

Having i apples seems similar in that it may sound like something that might sorta have some truth, inviting all sorts of brainstorming in trying to make sense of it.

Of course, it's gibberish, as are such "wise" expressions. But, just like malformed data can trick a computer into doing something that it wouldn't normally (obligatory xkcd), malformed expressions can trick a mind into acting erratically.

Mathematics is not truth.
Neither is it green, ten or auspicious.

Mathematics describes truth, or what is believed to be truth.
Some aspects of the truth it describes are not applicable to other aspects.
You can have 6 apples, and green apples, but not 6 greens*.
This is not a "fault" of mathematics.

*In that 'example' the failure is due to improper linguistic contruction, but the point is (hopefully) made.

"Smell the colour nine"

From a mathematician, not a philosopher:

We can use mathematics to model parts of reality. Part of what makes mathematics such a powerful tool is that the same mathematical construct can model many different physical phenomena, often in truly unexpected ways.

For instance, natural numbers can count discrete things (we have 3 apples). The positive real numbers can also measure "continuous" things (we have 3.14159 cups of water). The complex numbers do not seem to have an application to counting.

However, the real numbers can also be understood geometrically as scaling factors of images. You can think of a real number as a magnification factor for an image. So we can talk about scaling an image up by a factor of 3, or shrinking it by 1/3. Negative numbers make sense in this context, with some work: -1 represents a point reflection.

In this context, the imaginary number i corresponds to a 90 degree rotation of the image counterclockwise. Each complex number corresponds to a rotation and a scaling of the image. For instance the complex number 1+i corresponds to a rotation by 45 degrees and a scaling by sqrt(2) ~ 1.414.

So there is a physical application of these imaginary numbers. They just do not generalize counting.

Yes and no - the nature of truth is to some extent a matter of choice. Let me explain:

Yes: If a statement is derived, by means of mathematical logic, from a statement known to be true, then it is true.

No: At the very foundations of any mathematical theory, we make choices about what we regard as true without proof: these are the axioms. They are, in most cases, 'obviously true' (whatever that means), although there is at least one that is a bit uncomfortable: the axiom of choice. And in any case, we cannot prove that the axioms are in fact true in any other sense that we have chosen to consider them true.

That said - I think most people would find it very hard to argue against most of the fundamental axioms in maths, not because they are hard to comprehend, but because they match our intuitions and experience of reality so well.

Counting, the way you are used to it, isn't real. I mean, it makes lots of sense in our current low-entropy universe, where distinct things that are pretty similar are around, and we have lots of brains to notice it.

But 1 cookie, 2 cookies, 3 cookies -- that isn't a fundamental thing about our universe. There is a bunch of stuff. When ridiculously highly structured, as a shorthand, you can assign different stuff to have a similar label, and then count how many things have the same label. And there isn't only one kind of thing you can count, but more than one kind of thing! (Even more structure). And when you say two things have the same count, you can move them around (more entropy generated) and put them in correspondence to each other.

Counting is a common pattern in our low-entropy current universe epoch. Making the abstraction of counting -- 0, 1, 2, 3 and so on -- mostly lies on top of the fact that there are piles of things to count, and we know how counting things work.

Our universe's current state is ridiculously highly ordered because of how recent the big bang was. Entropy hasn't had time to grow to turn everything to a smear -- so while that is true, there are going to be patterns, and those patterns are going to be pretty similar to each other, and those similar patterns are going to be countable.

Once you say "there are 1 thousand sheep", and have an idea what people mean by "sheep", you can encompass a ridiculous amount of information really quickly. Given 1000 sheep, you know you could split them into pairs, and take the left and right halves, and each half would have 500 sheep.

Expressing that without counting would involve understanding and knowing each of the 1000 sheep as distinct things, individually talking about the concept of pairing and left/right for each of them, and then understanding each of the piles of things (called "sheep"). A real pain.

Mathematics acts as a kind of compression. We label things as sheep (a category of kinds of stuff clumped in a particular way), say we have 1000 different stuffs that can be labelled as them. That is much, much more structured than "we have 20,000 kg of various proteins, fats, minerals, liquids and carbohydrates arranged in this specific way". (Note I used a number there, hard to get around).

If you accept that -- that mathematics is compression, or shorthand, that lets you talk about patterns of various kinds in much cheaper ways -- then the rest of mathematics falls out.

What is $i=sqrt(-1)$? Why, it is yet another pattern. When you start with the counting numbers, you can then find the pattern of fractions. This pattern can be used to express things even more powerful than counting numbers.

From that you can find the pattern of the continuum -- the real numbers -- which again can be used to express even more powerful thoughts.

As it turns out, certain things can be expressed using polynomials; "x squared plus two x minus 3" for example. They are powerful tools that let you understand how things (in our highly ordered, low entropy universe) move, fall, and the like.

Those polynomials in turn are easier to work with if we invent a symbol we called "i", which when squared equals -1. It doesn't have to correspond to anything physical for it to be useful; in fact, in many situations its existence in a "solution" to a mathematical equation is strong evidence that there isn't a solution at all. But it merely existing makes finding the solution (or lack of it) easier; using the real numbers with "i" added (aka, the complex numbers) makes doing the mathematics (compression of understanding of reality) easier, and reality gets compressed better.

The mathematicians play the game of numbers carefully, and are pretty convincing that the addition of "i" doesn't break the game when played disconnected from counting.

So now we have these complex numbers. As it happens, you can find other parts of reality -- rotation, electrical potential, quantum mechanics and a whole pile of other things -- in which you can connect the complex numbers (including "i") to physical phenomena and patterns in ways that the complex numbers generate useful predictions of what happens next. They are good at compressing things. So they are useful mathematics (in the applied sense).

Are they "true"? Well, I'm starting from the position that counting isn't "true". They don't need to be "true" to express truth or be useful.

Because it is true that at least two yummy cookies are waiting for me at home, even if counting isn't really real.

Mathematics itself isn't truth, but all its results can be said to be true.

Everything in mathematics begins with a set of assumptions and definitions.

All proofs are pure deductive reasoning based on those assumptions and definitions. Every proof implicitly or explicitly begins with "Assuming A, B, and C are true, then … .".

There is no claim whatsoever that A, B, or C actually are true. The only claim is that if they are true, then the results of the mathematical proof must also be true.

Consider geometry, which begins with 5 postulates:

• A straight line segment may be drawn from any given point to any other.
• A straight line may be extended to any finite length.
• A circle may be described with any given point as its center and any distance as its radius.
• All right angles are congruent.
• Given a straight line and a point not on that line, there is exactly one other straight line that does not intersect the first.

There are thousands of theorems derived from these 5 simple assumptions, and there are other branches of mathematics based on them, such as trigonometry. All their results are undeniably true, but only if one assumes that the 5 postulates are also true.

The 5^th postulate looks like it could be proven from the first four, but in the thousands of years since Euclid proposed them, no one has ever been able to.

Meanwhile, other mathematicians wondered, what if we replace that last postulate with something else? Perhaps we can find contradictory proofs, and thereby prove that our different version of the postulate can't be true.

For instance, these two versions:

• Given a straight line and a point not on that line, there are no other straight lines that do not intersect the first.
• Given a straight line and a point not on that line, there are infinitely many other straight lines that do not intersect the first.

resulted in two branches of non-Euclidean geometry. And perhaps surprisingly, no contradictions within either mathematical system have ever been found.

So we have three completely different systems, with three completely different results, yet all three are "true" in the sense that all their results are true if the original postulates are true.

But no mathematician would ever claim that any postulate actually is true. Only that deductions based on the postulates must be true if the postulates are true.

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Back in the real world, one can see that many things look very similar to mathematical systems. For instance, if we look at something as small as a piece of paper, or as large as a field, we know we can mark straight lines and circles on them. And since the 5 postulates sound like they describe how the real world works, we can assume that any mathematical results derived from the 5 postulates will approximate how the real world operates.

And so we routinely use the results of Euclidean geometry when drawing on a piece of paper or surveying a field for building a new housing subdivision.

On a larger scale, the world isn't flat, so Euclidean geometry doesn't work very well. But, the surface of a sphere fits well into the non-Euclidean geometry in which there are no parallel lines. So now we can easily sail or fly across an ocean without getting lost.

Similarly, the field of physics uses mathematics as a tool to describe how the universe and everything in it works. And again, it is because the real world seems to approximately correspond to some basic mathematical postulates.

What is really interesting about this is that even though the approximations to reality of the results don't have to be correct, or even close to correct, it turns out that they are very close to correct. In fact, the universe seems to operate as if it were designed by a mathematician: the correspondence between theoretical results and measured reality is always perfect, within the precision with which we can make measurements.

Mathematics isn't truth, but in practice it seems to provide us with a very close approximation of reality.

I think it is a mistake to assume that there exists something like a context-independent notion of truth.

Let me explain what I mean with the context dependence of truth.

Consider the following simple question: Did Han shoot first?

Now you can observe that in the real world, as far as we can tell, Han didn't exist at all. Obviously a person that doesn't exist can't shoot, neither first nor second. So the obvious truth is: Han didn't shoot at all. Right?

But if you asked someone who has just seen the original showing of the film, and he answered that Han is not real and therefore didn't shoot, you would not be satisfied with the answer. Because the film clearly depicted the shot, and therefore anyone who has seen it should be able to answer the question. And the answer, in the context of the original version of the film, was: Yes, Han did shoot first. And anyone who claims otherwise either misremembers or is lying. So now, we get as obvious truth: Han shot first. And everyone who claims the opposite clearly is either mistaken, or is lying.

But what about people who have only seen the later, edited version of the film? Those will also have a clear answer to that question, and they all will agree: Han shot second. And anyone who claims differently is obviously lying.

So now we have three apparent truths which contradict each other: Han didn't shoot at all, Han shot first, and Han shot second. So which one is the actual truth?

Well, all three are the truth in their respective context. In the context of the real world, Han didn't shoot. In the context of the original version, Han shot first. And in the context of the edited version, Han shot second.

And we can clearly see that those are truths, because in each case, there's only one correct answer. Anyone who says that in the real world, Han shot first, is obviously not telling the truth. Anyone who says that in the original film, Han didn't shoot first, obviously doesn't tell the truth. Anyone who says that in the edited film, Han shot first, is not telling the truth.

So whether “Han shot first” is the truth depends on the context. It is a context-dependent truth.

And so are mathematical statements. The statement “there does not exist a number whose square is minus one” is true in the context of real numbers, false in the context of complex numbers, and meaningless in the context of the real world. There are no numbers in the real world, only things that can be described with numbers. Some things are better described by real numbers, and other things are better described by complex numbers. And some things aren't described well by either.

in mathematics there are purely imaginary numbers which cannot be represented directly in reality.

I think the latter part of this statement is invalid. Hence I question the validity of this question entirely. This part of the statement to be precise "which cannot be represented directly in reality". What defines reality? Do you mean that the physical world defines reality?

For example: Is a 3 dimensional equation less real because it cannot be painted on a 2 dimensional surface? I think it does not say anything about the equation/mathematics itself. The 3d outcome is not real on a 2d surface, that is for sure. The data to represent it is simply lost.

Another example. Is the mathematics of 3d depth of a 5d computer game (3d space + time + choice = 5d) not real because it is represented on a 2d screen? What is not real is that you do not perceive the depth, that part is not real. The data and mathematics to represent the depth are available though, you only need a 3d monitor to perceive it. The mathematics running on the computer stay exactly the same when you have a 3D or a 2D monitor. What changes is the way it is represented, but the mathematics remained the same.

tl;dr: Yes to pragmatists; no to everybody else: For them, mathematics is about correctness, not about truth.

While it is true that mathematics obviously was — and, perhaps less obviously, still is — inspired by our (perceived) reality, it is one of the essential traits of mathematics that it quickly and rigorously abstracts from that reality.³

In a very general way one could say that mathematics strives to find interesting statements about structures which are defined as tersely as possible, with minimalist definitions we call axioms. These trait definitions are unprovable because they are essentially arbitrary; but of course many of them, e.g. the Peano axioms describing the natural numbers, are inspired by reality. For example, our normal computation rules operating on natural numbers reflect the fact that in our macroscopic reality things do not, usually, spontaneously (dis)appear. But in the quantum realm things do (dis)appear, and suddenly we have to consider probabilities instead of discrete numbers.

Mathematicians take care to rigorously prove all their theorems. While one often calls proven theorems "true", a better term is "correct": What we mean is that the theorem follows necessarily from the given axioms. It does not contradict them. It is important to understand that this is just a statement about the realm defined by the axioms (natural numbers, real numbers, Euclidean space etc.); prima facie it is not a statement about our perceived reality.² (But more on that below.) This is why it does not even make sense to ask whether mathematical statements are "true": They are less then that and more than that at the same time: They are proven correct given the respective axioms. (This is less than true because it is a statement in and about a "mental sandbox"; but it is more because it is impossible to "prove" anything about "reality". All statements about reality, even those supported by the most rigorous scientific experiments, are (1) about the past; and (2) limited to the space, time and energy constraints under which they were performed.)

The emerging situation is a bit like a web of trust in cryptography: One can use previous mathematical theorems as building blocks for one's own thinking because they are proven correct. Anything correctly derived from previous theorems is therefore also proven correct.

Now let's finally examine the relation of math and reality. As mentioned in the beginning, many axioms and the structures defined by them are inspired by and have an obvious correspondence in our perceived reality: Natural numbers are used for counting, real numbers are used for measuring, even complex numbers are used: for electrical engineering. We call two systems which have a structural similarity homomorph. Operations in one system have an equivalent in the other system and lead to equivalent results. For example, there is a structural correspondence between real numbers and lengths, respectively areas. If we have two pieces of furniture of known width we do not have to place them next to each other to know how wide they are combined; we simply add the widths and know beforehand whether they'll fit along the wall in our living room. Likewise we'll know how much carpet we must buy for a room, without counting little squares on the floor: We simply multiply the lengths of the sides. Mathematical square roots and squares correspond to sides and square surfaces in our perceived reality.¹ The operation in the realm of real numbers is equivalent to an operation in reality because the definition of real numbers defines a system which is structurally similar to the physical reality — on small scales, and with small speeds: It breaks down with cosmologically long distances, large masses or high speeds relative to c.

In this sense we can make the following statement: Results obtained mathematically are (quite obviously) useful in reality. In a pragmatic sense one may call such results "true": Our furniture fits, our buildings don't collapse, and google's estimated travel times are usually not too far off.

But care must be taken not to exceed the area of structural similarity between the mathematical model and the reality we try to investigate. Unfortunately that boundary is unknown until it is crossed.

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_{¹ The astute reader will notice that because we live on the surface of a sphere the sides of a "square" are not "straight", and the surface area enclosed is not the square of their lengths — in fact, neither of the two enclosed surface areas is! One of them is just very close, as long as the square does not get too big. That's the "perceived" part. Of course we can fix that by using a non-Euclidean geometry. But if you measure very accurately you'll notice that space itself is constantly expanding and undergoes the occasional oscillation, not to mention Earth's gravity bending space time.}

_{² This statement is, of course, tongue-in-cheek the same way a movie claims "any resemblance to actual persons, living or dead, or actual events is purely coincidental" even though its characters, events and locations are readily recognizable.}

_{³ This abstraction is equally strength and weakness. It allows us to ignore the fuzzy and inscrutable convolutions of reality so that we can be sure of every aspect of what we talk about; but because it ignores most if reality is of limited use. "You must be a mathematician." - "Why?" - "What you say is 100% correct but completely useless."}

As an engineer, I would say that if something can be proven to be useful, then it is true in some sense. This philosophical stance that something being "true" is related to positive consequences or outcomes can be produced / derived from it has some specific fancy-pants Latin name which I knew when I was younger, but sadly seem to have forgotten now. ( Please fill me in if you know it. )

edit: This philosophy is called pragmatism and comes from greek language (thanks to Peter Schneider's comment)

Many branches of mathematics can be proven to be useful, although they are above the heads of most engineers and much above the rest of the population also.

An example is any kind of computer program taking an input and giving an output. The algorithm can use arbitrarily difficult and advanced mathematics. If we can agree that it gives a useful output for some well defined input, then the math being used is "true" in some sense.

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We can also decouple the whole reliance on engineering by instead asking ourself wether it makes sense to ask if an axiom should be considered true. Does it lead to an interesting or beautiful theory or not? Once again the same train of thought.

Does it lead to positive / interesting / useful consequences if we regard it as being true?

Ill formed question. Mathematics (specifically, logics) define what truth is. You are trying to test the validity of the tool with the tool itself. The answer would be a plain "yes". Otherwise (if you discuss mathematics as an issue of perception) you fall into Rusi's answer.

Yes, you can have i apples, if you define the domain of i (i is not just a parallel universe of numbers, it needs to be defined as a coherent domain). A quantity is just a mental idea, and ideas can't exist without a mind (check Locke/Berkeley). You can't have 10⁸² apples, even if such is a positive, real, natural, whole, integer number. Can you?

“You can’t have i apples”

As @Conifold points out you cannot even have √2 apples.

I'd go further.

Can you have -2 apples ⅓ apples?

I'd say (from a certain pov) no.

• All physics is based on measurements
• All measurements come from instruments
• Which can only ever deliver integral non-negative bounded multiples of least count

Note: I added the “bounded” aka finite above thanks to @RodolfoAP comment about impossibility of even 10⁸² apples.

So why is it that imaginary numbers seem to cause a special problem?

Maybe it's the word imaginary that is the problem?

This is...

More a linguistic relativity question

than we may realize.

Personal Experience with i

I must have been about 12 when a teacher gifted me George Gamov's 123 infinity ... my first brush with the "imaginary" numbers so-called. The whole book was fascinating, enthralling but I found that part much more incomprehensible than all the rest on (cantorean/hilbertean) infinity.

In adult retrospect the math of infinity is inherently harder (and IMHO more questionable) than the math of complex numbers.

What gives? (Or gave to that 12 year old)?

I conjecture it's...

The word imaginary

The word strongly, overwhelmingly conduces towards the suggestion of unreality.

But in retrospect we (mathematicians) could have chosen some completely different adjective-pair eg

• crooked-straight
• proper-improper
• green-red
  Note: How subatomic physicists chose "strangness" "charm" even "spin" in a whimsical way

And we would not have this question/confusion!
We may have others of course!

Despite some claims, the Cartesian myth that math is independent of physical reality is arguably false. Mathematics is NOT independent of the physical systems which embody it. Physical systems are structured in such a way that mathematical statements supervene on them. An excellent introduction into how mathematical truths are functions of conceptual mapping in the brain can be found in Lakoff and Nuñez's book Where Mathematics Comes From. It should be noted that to presume that an abstraction of math is independent of physical systems is based on one's metaphysics, and there's are a number of thinkers in the analytic empirical tradition (starting with Gilbert Ryle) that reject the duality and independence of mind and body. Following Lakoff and Nuñez's first collaboration is their philosophical work called Philosophy in the Flesh which details their attack on the common and historical philosophical assumptions starting with Plato and Aristotle which have seem to lost currency in the face of modern science.

in mathematics there are purely imaginary numbers which cannot be represented directly in reality. ex. you can't have i applies (i=square root of -1) can we then say that sometimes mathematics is not truth?

This is a good question to put you on a path to philosophy, so let's address the interrogative piecewise:

First, not everyone agrees on what truth is. Some believe that it is relationship between statements and how the world is (correspondence), others believe that is about the consistency of a statement with other statements (coherency), and yet other believe it is more about how a statement solves problems (pragmatism). So, whether or not mathematics is truth depends on your understanding of truth.

Now, the question you are putting forward is more along the lines of, if the natural numbers correspond to entities we can count, and those statements are true (I see 2 cookies on one plate, and two on another so there must be 4 altogether on the two plates). How does one make sense of the square root of a negative number? For instance, a square root is normally a number which when multiplied by itself gives a product called a square, and yet we know no number times itself can ever be negative. Certainly the first statement (2 + 2 = 4) seems to be true because it corresponds to the state of affairs regarding the cookies. So does that mean that sometimes mathematics doesn't correspond to physical reality, and therefore isn't really "true" in the correspondence sense? Absolutely. In this case, and many in mathematics, the truthiness of a statement seems lacking. Our intuition guides us in 2 + 2 = 4 to true, but steers us away when we say i := √-1. But remember the pragmatic theory of truth? It would argue that just because a number doesn't correspond with any state of affairs in the world doesn't make it false. Certainly i gets the job done! (Electrical engineering wouldn't really work without i in the modern sense, for instance.)

Here's an even better example that Lakoff and Nuñez cover in the book. It's one of Euler's equations, caled the Euler identity: e^πi = -1. What the heck are we supposed to make of raising an irrational to a product of an irrational and an imaginary and getting -1? How can this even be meaningful? It turns out that Euler's identity expresses a projection of a point on a line (a value in the domain of an angle) onto a circle (unit circle in the Trigonometric Complex Plane). (See page 439 for the geometrical projection that represents Euler's identity.) In other words, it's a fancy equation that expresses a simple geometrical truth! And geometry is how we understand space fundamentally. So the Euler Identity is a fancy truth about how we understand space-time. And nothing is more relevant to understanding physical reality than space time.

So just like 2 + 2 = 4 seems to correspond to physical reality, i and e^πi = -1 also correspond to physical reality. It's just harder to understand why. So, welcome to the philosophy of mathematics, and if you want to know about how mathematics can and cannot express truth, read up on truth!

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EDIT 2019-09-03

As per commentator's request, a clarification. No, not all mathematical assertions correspond to physical reality and cohere with each other depending on the context. In fact, we often create contradictory mathematical truths that may individually correspond to reality, but contradict each other to prove statements are incoherent with each other. Because physicality is a material cause of information structure doesn't mean that information structure, like a mathematical statement, must describe physicality; this is obvious in a non-mathematical information example. Horses gallop on the plains is perfectly true because it corresponds, coheres, and correctly functions as an assertion. Unicorns gallop on the plains is perfectly false because it does NOT do so. Mathematical statements are analogous.

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EDIT 2019-09-04 In regards to commentator's comments, no, one cannot choose ANY mathematical theorem. There are constraints on mathematical theorems. One cannot declare the value of Pi to the same number of decimal places as the atoms in the universe, because no mind could possibly hold that many places. This is a physical constraint, and shows that there is not a Platonic mathematical realm that floats independent of the universe, at least if one accepts the scientific method. Another constraint on theorems is semantic, another product of physical embodiment. One cannot use statements that cannot be understood by a brain. "Furgleflex plus sibblejibble equals jabjib" is simply not mathematical and would get one booted from a mathematical conference if one were to stand at a podium and declare it a mathematical theorem since it means nothing to the other mathematical brains in the room. A computer can generate even mathematically sounding theorems like "the corners of a circle have a quotient of 16 dimensions", and still it is not a viable mathematical theorem not because rocks or trees discriminate against it, but because human brains do. This is not some arbitrary coincidence. Mathematical semantics ultimately derives from the physical processes of the human brain alone. (NB Ethologists of course have shown that higher order animals all possess mathematical semantics to a much lesser degree.)