# Gottfried Leibniz, Carl Gauss and Neural Coding

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```					Gottfried Leibniz, Carl Gauss and
Neural Coding

Outline of talk to students of German language classes at
Bozeman High School
May 9, 2002
Gottfried Wilhelm Leibniz
1646-1716
Selected Chronology of Leibniz’s Life:
•   Born in Leipzig, Germany
Mom: Catherine Schmuck
•   Education
– At 14: attended University of Leipzig (1661)
– At 17: Degree (undergrad) in philosophy and mathematics (UL) (1663)
– Masters in Philosophy (UL) (1663 or 1664)
– At 20: Doctorate from University of Altdorf (1667)
•   Vocation: Diplomat for Germany traveling to the capitols of Europe.
– Discovered in 1676
– Published “A New Method for Maxima and Minima and also for tangents which is not
obstructed by Irrational Quantities” in 1684
– Controversy with Sir Isaac Newton (the other Daddy of Calculus) discovered the calculus in
1665 and published his version of it in his Philosophae Naturalis Mathematica in 1687.
– Profound discovery with much generality! Today it is used profusely in statistics, business,
economics, biology, sociology, psycology, engineering …
What is Calculus?
•   Calculus deals with extremely small (large) things, a concept that has eluded
mathematicians since the 6th century BC:
–   Pythagoreans (550 BC)
• Right triangles and the Pythagorean Theorem.
– 32 + 42 = 52!
– 32 + 52 = sqrt(34)2. Irrational quantities from square roots
• Pythagoreans believed that there was a fraction which was equal to sqrt(34). The fact is
that sqrt(34) is NOT a fraction, but you can get as (infinitesimally) close to sqrt(34) as
you’d like with some fraction.
• The ancient Greeks could not deal (rigorously) with the concept of the infinitesimal (or of
the infinite).
–   Paradoxes of Zeno (450 BC)
• Achilles and the Tortoise – a paradox of the infinitesimal!
•   Mr. J. drives to Sun Valley.
–   Distance curve
–   Steepness of the curve = how fast Mr. J. is deiving
–   Quantify this!! Steepness of curve = steepness (slope) of the tangent line to the curve
–   Where’s the infinitesimal? Ans: How do you get the tangent line?
•   Theme: Fractions get close to square roots, Achilles gets close to the tortoise, secants
get close to the tangent.
•   POINT: Calculus is about the steepness of tangent lines.
•   These kids know about simple polynomials like lines and parabolas
Johann Carl Friedrich Gauss
Prince of Mathematics
“Math is the queen of the sciences”
1777-1885
•   At 10, he summed 1, 2, 3, … 100 using n/2(n+1).
•   Education:
– At 11: attended Gymnasium funded by the Duke of Brunswick, Wolfenbottel
(1788)
– Degree (undergrad) at Brunswick Collegium Carolinum (1799)
– Doctorate at University of Helmstedt (1800-1801?)
• Fundamental Theorem of Algebra (thesis): A polynomial of degree n has n roots.
•   Vocation: director of Go..ttingen.
– Accurately predicted where Ceres, a new “small planet” (asteroid) discovered on
Jan 1, 1801 by Italian G. Piazzi, would reappear the next year.
•   Prince of Mathematics:
– Gaussian: unit of magnetic flux
– Gaussian Numbers (“complex integers”)
– Gauss’s Lemmas (Algebra) and Gauss’s Theorems (Complex) and Gaussian Plane
and Gaussian Distribution
Gaussian Distribution
• Weights of students in the class follow a
bell shaped curve!
• Lots of data follow this distribution!
• CLT
Neural Coding
• How does the nervous system communicate
– You are (maybe) looking and listening to me!
– Luke Skywalker gets his hand cut off in Empire Strikes
Back, Princess Lea rescues him, then a robot sews a
new hand on then pricks his prosthetic with a needle.
Luke says: “Ow!”
• This is possible only if the prosthetic can successfully
communicate pain (and heat and tactile info) with the nervous
system.
• How do the great German mathematicians Leibniz
and Gauss help neurobiologists try to answer this
question?
German Math Guys and Neural
Coding
• Leibniz’s Calculus helps us find where the
steepness of the curve is “flat”
– Like finding the vertex of a parabola
– These are Max’s and Min’s!
– This is precisely what we do with our model for neural
coding!
• Gaussian Distribution: Given a stimulus, there are
lots of responses of the nervous system. We
order the responses using Gauss’s bell shaped
curve!

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