Feynman: 'Greek' versus 'Babylonian' mathematics
Summary
TLDRThe script explores the nature of mathematics in physics, contrasting the Babylonian and Greek traditions. It emphasizes the interconnectedness of mathematical statements and the debate over which should be fundamental principles versus consequences. The discussion highlights the efficiency of the Babylonian method in physics and the broader validity of principles like the conservation of angular momentum, which extends beyond its original derivation in Newtonian mechanics to quantum mechanics, illustrating the importance of understanding the interconnections in physics.
Takeaways
- 🧩 Mathematics serves as a tool for reasoning and connecting various statements in physics.
- 🌐 The script discusses two traditions in mathematics: the Babylonian and the Greek, emphasizing their different approaches to learning and understanding.
- 📚 The Babylonian tradition focuses on learning through a multitude of examples to grasp general rules, while the Greek tradition emphasizes axioms and formal demonstrations.
- 📉 In Babylonian mathematics, theorems are interconnected without a strict foundational structure, allowing for flexibility in reasoning.
- 🏛 The Greek or modern mathematical approach starts with axioms and builds a structured framework of knowledge.
- 🔄 The script highlights the interconnectedness of mathematical theorems, suggesting that there isn't a single 'bottom' from which all else is derived.
- 🔍 The efficiency of mathematical methods is questioned, with the Babylonian method being more practical for physicists due to its flexibility.
- 🌌 The script uses the example of gravitation to illustrate the difference between fundamental principles and derived theorems in physics.
- 🌀 It discusses the conservation of angular momentum as a principle that extends beyond its initial derivation in physics, showing its wider applicability.
- 🚀 The script suggests that while physics may eventually reach a point where all laws are known and axiomatic reasoning is more straightforward, currently, physicists must balance various propositions and their relationships.
- 🔬 The importance of understanding the broader implications of physical laws, even when they extend beyond their original proofs, is emphasized for a complete understanding of physics.
Q & A
What is the fundamental concept of mathematics as described in the script?
-The script describes mathematics as a method of transitioning from one set of statements to another, which is useful for developing consequences, analyzing situations, and connecting various statements in fields like physics.
What are the two traditions of looking at mathematics mentioned in the script?
-The two traditions mentioned are the Babylonian tradition and the Greek tradition. The Babylonian tradition focuses on learning through a large number of examples to catch on to the general rule, while the Greek tradition emphasizes starting from axioms and building up the structure through logical deductions.
How does the Babylonian approach to mathematics differ from the Greek approach?
-The Babylonian approach involves learning through numerous examples and understanding various theorems and their connections without necessarily deriving everything from a set of axioms. In contrast, the Greek approach is more systematic, starting with axioms and using logical deductions to establish a framework of understanding.
What is an example of a theorem that could be seen as an axiom in one geometry perspective but not in another?
-The Pythagorean theorem can be seen as an axiom in the geometry perspective of Descartes, but in Euclid's geometry, it is derived from other axioms.
Why is the method of starting from axioms not always the most efficient way of obtaining theorems?
-Starting from axioms is not always efficient because it requires going back to the axioms each time to derive a theorem. It is more efficient to remember a few key principles and use them to navigate through the theorems and their interconnections.
What is the significance of the conservation of angular momentum in physics?
-The conservation of angular momentum is significant because it is a principle that remains valid across different laws and systems in physics, even when the original derivation (like Newtonian laws) is found to be incorrect or incomplete.
How does the script relate the conservation of angular momentum to the formation of spiral nebulae?
-The script suggests that as stars fall together to form a nebula, the conservation of angular momentum principle explains why they move slower when they are farther out and faster when they are closer in, leading to the qualitative shape of spiral nebulae.
What is the importance of understanding the interconnections of different branches of physics?
-Understanding the interconnections is important because laws often extend beyond the range of their deduction, and having a balance of various propositions and their relationships helps in making educated guesses and extending knowledge beyond what is currently proven.
Why is it necessary to have a balance between different mathematical or physical principles in one's understanding?
-A balance is necessary because it allows for a comprehensive understanding of the subject, enabling one to see how different principles interrelate and how they can be applied in various contexts, even when the laws are not fully known.
How does the script illustrate the concept of efficiency in learning and applying mathematical principles?
-The script illustrates efficiency by comparing the Babylonian method, which relies on memory and reconstruction of principles, to the Greek method, which is more systematic but potentially less efficient in practical application due to the need to refer back to axioms.
What is the implication of the script's discussion on the derivation of wide principles from specific laws in physics?
-The implication is that while specific laws can lead to the derivation of wide principles, these principles can sometimes be more universally valid than the original laws from which they were derived, suggesting that the principles can transcend the limitations of specific theories.
Outlines
📚 The Nature of Mathematics and its Traditions
This paragraph delves into the essence of mathematics as a method of transitioning between statements and its utility in physics for developing consequences and analyzing situations. It contrasts two historical mathematical traditions: the Babylonian, which focuses on learning through numerous examples to grasp general rules, and the Greek, which emphasizes axioms and structured demonstrations. The speaker ponders the possibility of a fundamental principle from which all mathematical truths can be deduced, or if there's an inherent order in nature that distinguishes between fundamental and consequential statements. The paragraph also touches on the interconnectedness of theorems and the efficiency of different approaches in learning and applying mathematics.
🌌 Axioms and Theorems in Physics and Astronomy
The second paragraph discusses the importance of choosing the right axioms in physics, using the example of gravitation to illustrate the point. It debates whether the force law or the principle of equal areas being swept in equal times should be considered a more fundamental axiom. The speaker introduces a generalized principle of conservation of angular momentum, which is applicable to systems with many interacting particles, such as celestial bodies. This principle is then used to explain the formation of spiral nebulae and the behavior of a skater spinning. The paragraph highlights the broader validity of physical principles beyond their original derivation, as seen in the conservation of angular momentum's applicability to quantum mechanics despite the inaccuracies in Newtonian laws.
🔍 The Interconnection and Extension of Physical Laws
In the final paragraph, the speaker emphasizes the need for a comprehensive understanding of the interconnections between various propositions in physics. It acknowledges that while the method of starting with axioms is not always the most efficient, it is crucial for grasping the broader implications of physical laws. The paragraph suggests that the validity of a principle is not solely dependent on its derivation but also on its experimental verification and applicability across different domains of physics. It concludes with the idea that a complete understanding of physics requires maintaining a balance between knowing the laws and recognizing their extensions beyond the scope of their original proof.
Mindmap
Keywords
💡Mathematics
💡Physics
💡Axioms
💡Deduction
💡Babylonian Tradition
💡Greek Tradition
💡Interconnection
💡Conservation of Angular Momentum
💡Quantitative Reasoning
💡Fundamental Principles
💡Spiral Nebulae
Highlights
Mathematics as a method for transitioning between different sets of statements and its utility in physics.
The interconnectedness of various mathematical statements and the minimal knowledge required by a physicist.
Introduction of the Babylonian and Greek traditions in mathematics, contrasting their approaches to learning and understanding.
The Babylonian method of learning through numerous examples to grasp general rules.
The Greek method of starting with axioms and building a structured framework of knowledge.
The debate on the efficiency of starting with axioms versus knowing various theorems in geometry.
The concept that theorems in mathematics are interconnected and can be derived from different starting points.
The importance of choosing the right axioms in modern mathematics for efficient theorem derivation.
The inefficiency of the axiomatic method in obtaining theorems compared to knowing a few key principles.
The necessity of the Babylonian method in physics over the Greek method for practical problem-solving.
The dilemma of choosing between fundamental principles like the force law versus the area swept in equal times in gravitation.
The generalization of the principle of equal areas swept in equal times to a broader conservation law.
The application of the conservation of angular momentum in understanding the formation of spiral nebulae.
The analogy between the skater spinning and the conservation of angular momentum in celestial mechanics.
The challenge of deducing principles that extend beyond their original derivation in physics.
The discovery that conservation laws in physics, such as angular momentum, are more universally valid than initially thought.
The philosophical problem of balancing the need for axioms with the recognition of their limitations in understanding physics.
The importance of maintaining a comprehensive understanding of the interrelationships between various propositions in physics.
Transcripts
of
mathematics mathematics then is a way of
going going from one set of statements
to another it's evidently useful in
physics because we have all these
different uh ways that we can speak of
things and it permits us to develop
consequences and analyze the situations
and rechange the laws in different ways
and to connect all the various
statements so that as a matter of fact
the total amount that a physicist knows
is very little he has only to remember
the rules for getting from one place to
another and he's able to do that do it
then in other words all of the various
statements about equal times of forces
in a direction of the radius and so on
are all interconnected by reasoning now
an interesting question comes up is
there some pattern to it is there a
place to begin fundamental principles
and deduce the whole
works or is there some particular
pattern or order in nature in which we
can understand that these are more
fundamental statements and these are
more consequential
statements there are two kinds of ways
of looking at mathematics which for the
purpose of this lecture I will call the
Babylonian tradition and the Greek
tradition in Babylonian schools in
mathematics the student would learn
something by doing a large number of
examples until he caught on to the
general
rule also a large amount of geog
geometry for example was known a lot of
properties of circles theorem of
Pythagoras for example formulas for the
areas of cubes and triangles and
everything else and some degree of
argument was available to go from one
thing to
another tables of uh numerical
quantities were available so that you
could solve elaborate equations and so
on uh everything was prepared for
calculating things out but uid
discovered that there was a way in which
all of the theorems of geometry could be
ordered from a set of axians that were
particularly simple and you're all
familiar with that much geometry I'm
sure
but the Babylonian attitude was if I may
my my way of talking what I call
Babylonian mathematics is that you know
all these various theorems and many of
the connections in between but you've
never really realized that it could all
come up from a bunch of
acents modern mathematics the most
modern mathematics concentrates on
axioms and demonstrations within a very
definite framework of
conventions of what's acceptable and not
acceptable as axioms for example in
Geometry it take something like nucle
zums modified to be made more perfect
and then to show the deduction of the
system for instance it would not be
expected that a theorem like Pythagoras
is that the sum of the squares of the
areas of squares put on the sides of the
triangle will equal the area of a square
on a hypotenuse should be an axum on the
other hand from another point of view of
ma of geometry that of decart the
Pythagorean theorem is an axium so the
first thing we have to worry about is
that even in
mathematics you can start in different
places because of all these various
theorems are interconnected by reasoning
there isn't any real way to say well
these on the bottom here are the bottom
and these are connected through logic
because if you were told this one
instead or this one you could also run
the logic the other way if you weren't
told that one and work out that one it's
like a bridge with lots of me members
and it's overc connected if pieces have
dropped out you're can reconnect it
another
way the mathematical tradition of today
is to start with some particular ones
which are chosen by some kind of
convention to be aums and then to build
up the structure from there the
Babylonian thing that I'm talking about
which I don't really not Babylonian but
it's is to say well I know happen to
know this and I happen to know that and
maybe I know that and I work everything
out from there then next tomorrow I
forgot that this was true but I
remembered that this was true and then I
reconstructed again and so on I'm never
quite sure of where I'm supposed to
begin and where I'm supposed to end I
just remember enough all the time so
that as the memory Fades and the pieces
fall out I reput the thing back together
again every
day the method of starting from the
axioms is not efficient in obtaining the
theorems in working something out in
Geometry you're not very efficient if
each time you have to start back at the
ax but if you have to remember a few
things in the geometry you can always
get somewhere else it's much more
efficient to do it the other way and the
what the best axum are are not exactly
the same in fact are not ever the same
as the most efficient way of getting
around in the
territory in physics we need the
Babylonian method and not the the uh
ukian or Greek method and I would like
to say
why the problem in the ukian method is
to make something about the axioms a
little bit more interesting or important
but the the question that we have is in
the case of gravitation is it more
important is it more basic is it more
fundamental is it a better axium to say
that the force is directed toward the
Sun or to say that equal areas are swept
in equal
time well from one point of view the
forces is better because if I State what
the forces are I can deal with a system
with many particles in which the orbits
are no longer ellipses because of the
pull of one on the other and the theorem
about equal areas fails therefore I
think that the force law ought to be an
axium instead of the
other on the other hand the principle
that equal times are swept out and equal
equal areas are swept out in equal times
can be generalized when there's a system
of a large number of particles to
another theorem which I had prepared to
explain but I see I'm running out of
time but there's another statement which
is a little more General than equal
areas and equal times well I have to
State what it is it's rather complicated
to say and it's not quite as pretty as
this one but it's it's obviously the the
sun of this one I mean it's The
Offspring if you look at all these
particles Jupiter Saturn the Sun and all
these things going around lots of stars
or whatever they are all interacting
with each other and look at it from far
away and project it on a plane like this
picture then everything everything is
moving this moving this way and moving
that way and so on then take any point
at all say this point and then calculate
how much each one is changing its area
how much area is being swept out by the
radius to every par ofle and add them
all together but wait those masses which
are heav
count more strongly if this is twice as
heavy as this one then this area counts
twice as much
so that's doing the sweeping and the
total of all of that is not changing in
time that's the generalization obviously
of the other one incidentally the total
of that is called the angular momentum
and this is called the law of
conservation of angular momentum
conservation just means that it doesn't
change now one of the consequences of
this is just to show what it's good for
imagine a lot of stars falling together
to form a nebula a
Galaxy as they come closer in if they
were very far out and moving slowly so
there was a little bit of area being
generated but on very long arm
distances from the center then if the
thing falls in the distances to the
center are shorter now if all the stars
are now close in the these radi are
smaller and in order to sweep out the
same area they have to go a lot faster
so as the things come in they swing
swirl around and thus we can roughly
understand the qualitative shape of the
Spiral nebula can also understand in the
same way exactly the same way way a
skater spins when you start with her leg
out uh it's moving slowly and as you
pull the leg in it spins faster because
when the leg is out it's contributing
when it's moving slowly a certain amount
of area per second and then when it
comes in to get the same area you have
to go around
faster but I didn't prove it for the
skater the skater uses muscle Force
gravity is is a different Force yet it's
true for the
skater now we have a problem we can
deduce often from one part of physics
like the law of gravitation a principle
which turns out to be much more valid
than the derivation this doesn't happen
in mathematics that the theorems come
out in places where they're not supposed
to
be in other words if we were to say that
the postulates of physics were the slow
of gravitation we could deduce the
conservation of angular momentum but
only for gravitation but we discover
experimentally that the conservation of
angular momentum is a much wider thing
now Newton had other pipe postulates by
which he could get the more General
conservation law of angular momentum but
Newtonian laws were wrong there's no
forces it's all a lot of baloney the
particles don't have orbits and so on
yet the analog the exact transformation
of this principle about the areas the
conservation of angul momentum is true
with atomic motions in quantum mechanics
and is still as far as we can tell today
exact so we have these wide principles
which sweep across all the different
laws and if one takes too seriously his
derivations and feels that this is only
valid because this is valid you cannot
understand the interconnections of the
different branches of physics someday
when physics is complete then maybe with
this kind of argument we know all the
laws then we could start with some
axioms and no doubt somebody will figure
out a particular way of doing it and
then all the all the deduction will be
made but while we don't know all the
laws we can use some to make guesses at
theorems which extend beyond
the
proof so in order to understand the
physics one must always have a neat
balance and contain in his head all of
the various propositions and their inter
relationships because the laws often
extend beyond the range of their
deduction this will only have no
importance when all the laws are known
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