Euler's Identity is generally expressed by the simple equation
e^(i*pi) + 1 = 0.
There is a beautiful illustrated account of Euler's Identity (and its proof) to be found on YouTube.
Perhaps more than any other relationship in Mathematics this requires a Type 2 mathematical interpretation.
Indeed indirectly the limitations of the conventional Type 1 approach in revealing the true nature of the Euler Identity is highlighted in the well known comment of Benjamin Pierce (quoted at the beginning of the clip):
"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it and we don't know what it means. But we have proved it and therefore we must know that it is the truth."
The fact that it is absolutely paradoxical from a Type 1 perspective based on linear reason, then this clearly points to the need for the alternative (Type 2) circular logical explanation.
Also the admission that "we don't know what it means" clearly indicates that a qualitative holistic interpretation (that cannot be provided by Type 1 Mathematics) is required.
The clip end with another interesting quote from Keith Devlin:
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the very beauty of the human form that is far more than skin deep, Euler's equation reaches down into the very depths of existence."
Once again this beautiful comment clearly hints at a meaning to Euler's Identity that greatly transcends conventional Type 1 interpretation. So there is a deep unconscious significance to the Identity, which Type 1 Mathematics is powerless to explore.
Now the first step to appreciating the true significance of Euler's Identity is to rewrite it with a very simple transformation.
So if e^(i*pi) + 1 = 0, then e^(i*pi) = - 1.
Then squaring both sides we obtain the more fundamental relationship,
e^(2*i*pi) = 1 (which I refer as the fundamental Euler Identity).
Now I have already in previous contributions pointed to the qualitative significance of these key mathematical symbols.
So e is qualitatively unique in the manner that it seamlessly combines both linear (discrete) and circular (continuous) aspects in its own identity.
Then i - from a qualitative perspective - represents an indirect rational means of conveying holistic type meaning in a linear quantitative context.
Pi which in quantitative terms represents the relationship as between the circular circumference and its line diameter again in qualitative terms points to a perfectly harmonious balance as between linear and circular type understanding.
Now as we have seen in conventional Type 1 terms, numbers when used to represent dimensions are treated in a merely reduced quantitative manner.
The key qualitative significance of the fundamental Euler Identity is that the dimensional expression i.e. 2*i*pi needs be equally interpreted from both a quantitative (Type 1) and (qualitative) Type 2 perspective.
Now I must admit that I have spent several years trying to precisely articulate the nature of the following conundrum:
as both e^0 = 1 and e^(2*i*pi) = 1, does this not imply that 2*i*pi = 0?
Well, clearly this is unsatisfactory from a conventional quantitative perspective. For it would follow from this acceptance that when we square both sides that - 4*(pi)^2 = 0 which is untenable as - 4*(pi)^2 represents a real magnitude!
However, from a qualitative (Type 2) perspective, 2*i*pi is indeed = 0!
One can perhaps begin to appreciate this fact by considering the circumference of a circle = 2*pi*r. Now in the conventional quantitative sense, where the circle is of radius = 1, then this gives an answer = 2*pi. However if we were to somehow conceive of the circle having an imaginary (rather than real) radius - which befits qualitative consideration - then the circumference would indeed be 2*pi*i (i.e. 2*i*pi).
So 2*i*pi in this context clearly points to a pure circular notion of dimension.
Just as the (reduced) linear quantitative notion of dimension is appropriate to Type 1, the circular qualitative notion is by contrast properly appropriate to Type 2 mathematical understanding.
Thus the key paradox with respect to the fundamental Euler identity relates to the fact that though expressed in a quantitative manner its true significance is of a qualitative (Type 2) rather than quantitative (Type 1) nature.
So by distinguishing the quantitative and qualitative type considerations, we are able to avoid the inappropriate quantitative conclusion that 0 = 2*i*pi.
Put another way e^0 = 1 is the quantitative (Type 1) expression of which e^(2^i^pi) = 1 is the corresponding qualitative (Type 2) equivalent. So, when we attempt to represent the pure qualitative (circular) expression for a dimension i.e. 2*i*pi in reduced (linear) quantitative terms it appears - quite literally - as 0!
Indeed there is an even simpler - and equally puzzling - expression that needs clarification.
In Type 1 terms both (- 1)^2 and (+ 1)^2 = 1.
Therefore we might be tempted to erroneously conclude that - 1 = + 1!
In fact this is a problem of the first magnitude that is not dealt with at all in conventional Type 1 terms!
It can only be resolved by introducing Type 2 notions.
So properly expressed from a Type 2 perspective (- 1)^2 = 1^1 whereas (+ 1)^2 = 1^2.
And in qualitative (Type 2) mathematical terms, 1^1 and 1^2 are distinct expressions.
However if mathematicians used the Type 1 approach consistently then they should be concluding by their own logic that - 1 = + 1!
Now this is issue is fudged in practice by confining calculations to the "principle" root, though strictly this is not the correct root.
So for example the true square root of 1 is - 1 (and not + 1).
Remarkably as we shall see this conclusion is supported by the extended use of the fundamental Euler Identity.