That mathematics can be done without these worries doesn't matter.

There is nothing wrong with that.

There seem to me to be several problems with such a principle.

points out that what this means is that such statements must be true because of their meaning, since they cannot be made true by anything in the formal system.

Since  is the solution to x2= -1, then it is as "real" as any other number.

I do not see why that must be the case.

Furthermore, if the complex number plane exhausted the meaning of imaginaries, then imaginaries would be completely redundant: all we would need is an extra spatial axis, which can be handled just fine without imaginaries.

That would be true if it is necessarily the case that some number must solve the expression -1.

When research on environmental problems seems like it is not having enough of an impact, mature idealists turn to outreach. This is convenient from the standpoint of career advancement because academics are expected to engage the community. Advising the local chapter of , giving a talk at a local science museum, and serving on a government advisory panel are all counted by promotion committees. More often than not, the combination of meaningful research, mentorship and a few hours per week of outreach fulfills the need of the researcher to improve the planet. But encouraging forays into the real world comes with unintended consequences as researchers are exposed to situations where the system designed to protect public health and the environment has failed. Facing injustice, an idealistic researcher might just step over the imaginary line that separates the dispassionate researcher from the environmental activist.

Just because a mathematical operation is conceivable doesn't mean that it must produce a solution.

This is a circular and question-begging procedure.

The effect of this, which elevates the subject to something nearing an equal partner with objects in reality, is to dignify non-existent objects, such as imaginary numbers, with greater reality than they would seem to possess otherwise.

We should listen to the logicians quietly snorting in the background.

And, once the question about meaning is seriously asked, it is not surprising, as in any question about , that philosophy and metaphysics should be involved.

The expression "" consists of a variable for an unknown.

But, however novel in science, this effect of the subject on objective reality is nothing new in philosophy: already saw in experience as the result of an interaction between external and internal, which makes it possible to speak of a of quantum mechanics.

Its only property is that, when squared, we get -1.

The idea has merit in reducing properties of complex numbers to properties of reals -- and thus avoiding the mysterious -1 -- but it does not tell or suggest anything new about complex numbers.

It is numbers that are foundational, not equations.

The place of observation in quantum mechanics has stood up under every experimental test that has been devised to challenge it, including ones ultimately suggested by Einstein himself.

Imaginaries are just one way we've decided to do it.

However, we have , in effect, the move of understanding imaginaries as pairs of real numbers; for the use of the "complex number plane" is no less than to substitute something real, the plane, for the meaning of -1, something which, as it happens, is governed by an ordered pair of .

Don't bother reading the essay.

This may be a bit too much in the way of metaphysics just for us to get imaginary numbers -- it might make it sound like mathematicians really must be "mystics" after all, as the professor of Asimov's anecdote said -- but there are other possibilities.