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
Daddy of Calculus
1646-1716
Selected Chronology of Leibniz’s Life:
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Born in Leipzig, Germany
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Dad:Friedrich
Mom: Catherine Schmuck
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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)
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Vocation: Diplomat for Germany traveling to the capitols of Europe.
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Daddy of Calculus
– 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?
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Calculus deals with extremely small (large) things, a concept that has eluded
mathematicians since the 6th century BC:
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Mr. J. drives to Sun Valley.
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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!
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
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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.
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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.
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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
information about the environment?
– 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
• Start with a model (like a line or parabola!)
• 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!