Feynman: 'Greek' versus 'Babylonian' mathematics

Summary

The 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

00:00

📚 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.

05:02

🌌 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.

10:04

🔍 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.

Keywords

💡Mathematics

Mathematics is a field of study that deals with the properties and relationships of numbers, quantities, shapes, and patterns. In the video, it is presented as a critical tool for understanding and analyzing the physical world, allowing for the derivation of consequences and the interconnection of various statements about the world.

💡Physics

Physics is the natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. The script mentions physics as a discipline that heavily relies on mathematics to describe and understand the laws of nature.

💡Axioms

Axioms are fundamental starting points or assumptions that are accepted without proof in a given context. In the script, they are discussed as the basis from which mathematical and physical theories are built, with the example of Euclid's modified axioms in geometry.

💡Deduction

Deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. The video discusses deduction in the context of deriving mathematical theorems from axioms and the broader implications for understanding physical laws.

💡Babylonian Tradition

The Babylonian tradition in mathematics, as described in the script, refers to a learning approach that involves working through numerous examples to grasp general rules, rather than starting from axioms. This tradition is contrasted with the Greek tradition, which emphasizes axiomatic systems.

💡Greek Tradition

The Greek tradition in mathematics, as mentioned in the video, is an approach that focuses on building mathematical structures from a set of axioms. This method is exemplified by the works of Euclid and is contrasted with the more empirical Babylonian tradition.

💡Interconnection

Interconnection refers to the linking or networking of various elements. In the context of the video, it is used to describe how different mathematical statements and physical laws are related through reasoning, allowing for a comprehensive understanding of complex systems.

💡Conservation of Angular Momentum

Conservation of angular momentum is a fundamental principle in physics stating that the total angular momentum of a closed system remains constant unless acted upon by an external torque. The script uses this principle to illustrate a broader point about the generalization of physical laws and their application to various phenomena, such as the formation of spiral nebulae.

💡Quantitative Reasoning

Quantitative reasoning involves the use of mathematical methods to analyze and solve problems. The video discusses how quantitative reasoning is essential for physicists to understand and predict the behavior of physical systems, such as the motion of celestial bodies.

💡Fundamental Principles

Fundamental principles are the basic theories or laws that form the foundation of a field of study. The script raises the question of whether there is a set of fundamental principles from which all of physics can be deduced, or if the nature of the subject matter requires a more flexible approach.

💡Spiral Nebulae

Spiral nebulae, as mentioned in the script, are a type of astronomical object characterized by a spiral structure. The video uses the conservation of angular momentum to explain the formation and structure of spiral nebulae, demonstrating how physical laws can be applied to understand cosmic phenomena.

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