> WYV o#bjbj 4:o48,q0d@@@@$S R@@+v@@|@*HA0qP, 7: || Lqq : Spring 2018 Professor Z. Damnjanovic
M, W, F 9-9:50, SOS B37 (213) 740-4084
Office hours: M, W, F 8-9 E-mail: zlatan@usc.edu
& by appointment
GESM 160g: Seminar in Quantitative Reasoning
Taming the Infinite: How Math Came to Be
The topic of the seminar is the Greeks invention of mathematics as a logical theory, with special focus on the evolution of the idea of mathematical infinity, from the Pythagoreans to Archimedes. This is an area of interface among mathematics, physics and philosophy.
In developing conceptual tools and computational techniques for solving various practical and physical problems, mathematicians very early on confronted a fundamental difficulty: how to reduce reasoning about continuous, spatially extended magnitudes such as area and volume to numbers-- aggregates of discrete, atom-like units that can be counted. Attempts to solve the problem directly seemed to lead straight into paradox and contradictions. Sophisticated strategies evolved to bridge the gap between the continuous and the discrete, the geometric and the arithmetical, through construction of the new systems of irrational numbers-- numbers determinable only by means of an infinite process -- and infinitesimal numbers -- numbers smaller than any finite positive number yet not equal to zero. Unable to do without them, and trying to justify reasoning about them, mathematicians found themselves unwittingly and unwillingly ensnared in problems logical, philosophical, physical and metaphysical. We follow the struggle against these difficulties, learning the ever more ingenious methods of reasoning invented to tackle them.
No knowledge of any particular branch of mathematics (including calculus) is presupposed. Lectures assume only high-school algebra.
Tentative list of topics covered:
early computational mathematics: finding square roots and greatest common divisors
Egyptian mathematics: computing areas and volumes by cut-and-paste
the Pythagorean theory of number
geometric algebra
the paradoxes of Zeno: are there atoms of space and time
the three classical problems: squaring the circle, doubling the cube, and trisecting an angle
the discovery of incommensurability
axiomatization of geometry: why and how
Eudoxean theory of proportion
- the method of exhaustion: Euclid and Archimedes
- Archimedess mechanical method and measurement of the circle: reasoning with the infinite
Requirements: 2 (two) essay type papers, 8-10 pages long, on pre-assigned topics; occasional problem assignments and classroom presentations.
Readings:
Euclid, The Elements
Artmann, Euclid: The Creation of Mathematics
Stein, Archimedes: What did he do besides cry Eureka
Katz, A History of Mathematics: An Introduction, 3rd ed.
Additional readings will be posted on Blackboard.
Statement on Academic Conduct and Support Systems
Academic Conduct:
Plagiarism presenting someone elses ideas as your own, either verbatim or recast in your own words is a serious academic offense with serious consequences. Please familiarize yourself with the discussion of plagiarism inSCampusin Part B, Section 11, Behavior Violating University Standards HYPERLINK "https://policy.usc.edu/scampus-part-b/" \t "_blank" policy.usc.edu/scampus-part-b. Other forms of academic dishonesty are equally unacceptable. See additional information inSCampusand university policies on scientific misconduct,http://policy.usc.edu/scientific-misconduct.
Support Systems:
Student Counseling Services (SCS) (213) 740-7711 24/7 on call
Free and confidential mental health treatment for students, including short-term psychotherapy, group counseling, stress fitness workshops, and crisis intervention. HYPERLINK "https://engemannshc.usc.edu/counseling" \t "_blank" engemannshc.usc.edu/counseling
National Suicide Prevention Lifeline 1 (800) 273-8255
Provides free and confidential emotional support to people in suicidal crisis or emotional distress 24 hours a day, 7 days a week. HYPERLINK "https://urldefense.proofpoint.com/v2/url?u=http-3A__www.suicidepreventionlifeline.org_&d=DwMFAg&c=clK7kQUTWtAVEOVIgvi0NU5BOUHhpN0H8p7CSfnc_gI&r=CkL4kY0no88QZj2uCDFOI7xhjLQ9O4YZiOIPGjQSQJ8&m=FN8fbuOqtif1bWvrKTHrycKDSgd_snFubXhYrp7Dz58&s=yigxVd9JJfatu9uXYx0qpj6LhjknPiz6tnBwI27pj4U&e=" \t "_blank" www.suicidepreventionlifeline.org
Relationship and Sexual Violence Prevention Services (RSVP) (213) 740-4900 24/7 on call
Free and confidential therapy services, workshops, and training for situations related to gender-based harm. HYPERLINK "https://engemannshc.usc.edu/rsvp/" \t "_blank" engemannshc.usc.edu/rsvp
Sexual Assault Resource Center
For more information about how to get help or help a survivor, rights, reporting options, and additional resources, visit the website: HYPERLINK "http://sarc.usc.edu/" \t "_blank" sarc.usc.edu
Office of Equity and Diversity (OED)/Title IX Compliance (213) 740-5086
Works with faculty, staff, visitors, applicants, and students around issues of protected class. HYPERLINK "http://equity.usc.edu/" \t "_blank" equity.usc.edu
Bias Assessment Response and Support
Incidents of bias, hate crimes and microaggressions need to be reported allowing for appropriate investigation and response. HYPERLINK "https://studentaffairs.usc.edu/bias-assessment-response-support/" \t "_blank" studentaffairs.usc.edu/bias-assessment-response-support
The Office of Disability Services and Programs
Provides certification for students with disabilities and helps arrange relevant accommodations. HYPERLINK "http://dsp.usc.edu/" \t "_blank" dsp.usc.edu
Student Support and Advocacy (213) 821-4710
Assists students and families in resolving complex issues adversely affecting their success as a student EX: personal, financial, and academic. HYPERLINK "https://studentaffairs.usc.edu/ssa/" \t "_blank" studentaffairs.usc.edu/ssa
Diversity at USC
Information on events, programs and training, the Diversity Task Force (including representatives for each school), chronology, participation, and various resources for students. HYPERLINK "https://diversity.usc.edu/" \t "_blank" diversity.usc.edu
USC Emergency Information
Provides safety and other updates, including ways in which instruction will be continued if an officially declared emergency makes travel to campus infeasible. HYPERLINK "http://emergency.usc.edu/" \t "_blank" emergency.usc.edu
USC Department of Public SafetyUPC: (213) 740-4321 HSC: (323) 442-1000 24-hour emergency or to report a crime.
Provides overall safety to USC community. HYPERLINK "http://dps.usc.edu/" \t "_blank" dps.usc.edu
t}~i y }
/0dqzúˋhv6CJaJhWh/f'6CJaJhh/f'6CJaJhh/f'CJaJh"h"6CJaJh"6CJaJh"CJaJh/f'6CJaJhvCJaJh/f'h/f'CJaJhjCJaJh]e~CJaJhwCJaJ0tH Y Z [
$
%
56/0RSX
&Fgd/f'gd/f'gdw WMNOZ[q
@ABtv-DM
]^gd
-DM
gdgdwgd/f'<^<gd/f'
&Fgd/f'
?@Bstuvjqqyϱ}nVVEV} hh>*B*OJQJph.jhhB*CJOJQJUaJphhhB*OJQJph#hh6B*OJQJ]ph"""hhB*OJQJph"""%hhB*CJOJQJaJph+hh5B*CJOJQJ\aJph"""h/f'CJaJhwCJaJhWh/f'6hWh/f'6CJaJhvCJaJhvhvCJH*aJv/68:rMOnpSUAChy{GIgd
-DM
gd./45789:qr)*KLMNO웃r`LLrL`&jhhB*OJQJUph#hh5B*OJQJ\ph hh>*B*OJQJph.jhhB*CJOJQJUaJphhhB*OJQJph#hh6B*OJQJ]ph+hh5B*CJOJQJ\aJphhhB*OJQJph"""hhCJaJ%hhB*CJOJQJaJphOSTlmnopDEQRSTU01?@ABCgh?@wxy˳ڳ˳ڳڐ˳ڳz˳ڳ+hh5B*CJOJQJ\aJph#hh5B*OJQJ\ph hh>*B*OJQJph.jhhB*CJOJQJUaJphhhB*OJQJph%hhB*CJOJQJaJph#hh6B*OJQJ]ph*yz{9:EFGHIvw E F ` a b c d t u (!)!]!^!o!p!q!r!s!!!ݼݤݼݼݤ݀n_ݤݼhhB*OJQJph"""#hh6B*OJQJ]ph"""%hhB*CJOJQJaJph""" hh>*B*OJQJph.jhhB*CJOJQJUaJphhhB*OJQJph#hh6B*OJQJ]ph%hhB*CJOJQJaJphhhB*OJQJphU%Iwb d u q!s!!u"w""T#V#W#X#Y#Z#[#\#]#^#_#`#a#b#$dd@&[$\$a$gd1
-DM
gdj
-DM
gd!-"."a"b"s"t"u"v"w"""""##F#G#R#S#V#W#^#b#o#شŢشpbWhj5CJ\aJhjhj5CJ\aJh5B*CJ\aJph""" hhB*CJ\aJph"""#hh6B*OJQJ]ph"""#hh6B*OJQJ]ph hh>*B*OJQJph%hhB*CJOJQJaJph.jhhB*CJOJQJUaJphhhB*OJQJphb#c#d#e#f#g#h#i#j#k#l#m#n#o#$dd@&[$\$a$gd1
21h:pD,/ =!"#$%j666666666vvvvvvvvv666666>6666666666666666666666666666666666666666666666666hH6666666666666666666666666666666666666666666666666666666666666666662 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~ OJPJQJ_HmH nH sH tH D`DwNormalOJPJQJ_HmH sH tH DA`D
Default Paragraph FontRiR0Table Normal4
l4a(k (
0No ListLoL
x_x_msonormaldd[$\$CJaJ4U`4 0 Hyperlink >*phJ^`J 0Normal (Web)dd[$\$CJaJ8o!8x_x_msohyperlinkPK![Content_Types].xmlN0EH-J@%ǎǢ|ș$زULTB l,3;rØJB+$G]7O٭V$!)O^rC$y@/yH*)UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f
W+Ն7`gȘJj|h(KD-
dXiJ؇(x$(:;˹!I_TS1?E??ZBΪmU/?~xY'y5g&/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ
x}rxwr:\TZaG*y8IjbRc|XŻǿI
u3KGnD1NIBs
RuK>V.EL+M2#'fi~Vvl{u8zH
*:(W☕
~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4=3N)cbJ
uV4(Tn
7_?m-ٛ{UBwznʜ"ZxJZp;{/<P;,)''KQk5qpN8KGbe
Sd̛\17 pa>SR!
3K4'+rzQ
TTIIvt]Kc⫲K#v5+|D~O@%\w_nN[L9KqgVhn
R!y+Un;*&/HrT >>\
t=.Tġ
S; Z~!P9giCڧ!# B,;X=ۻ,I2UWV9$lk=Aj;{AP79|s*Y;̠[MCۿhf]o{oY=1kyVV5E8Vk+֜\80X4D)!!?*|fv
u"xA@T_q64)kڬuV7t'%;i9s9x,ڎ-45xd8?ǘd/Y|t&LILJ`& -Gt/PK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧60_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!0C)theme/theme/theme1.xmlPK-!
ѐ' theme/theme/_rels/themeManager.xml.relsPK]
o!:Oy!o#vIb#o#
4)KSlDQ0??w9EE`(]o-asFRoXXXXXXXXXXXX8@0(
B
S ?hs
]
i
qxj
q
qx]`q&
)
agrwu'q3333333333333
G^q
22
q?@@BqOhz9ڊ^`OJPJQJ^Jo(hH-
^`hH.
pp^p`hH.
@@^@`hH.
^`hH.
^`hH.
^`hH.
^`hH.
PP^P`hH.Oh Oh @Rb9T)#0XbS,Lt &E
w/f'D,wFvux]e~1>j"]2oq@^^^^o@UnknownG*Ax Times New Roman5Symbol3.*CxArial7.@CalibriA$BCambria Math"qhZ:a W1W1!20aa3HP $Pw2!xxZlatanShared AccountOh+'0=
,
8DLT\dZlatanNormalShared Account4Microsoft Office Word@vA@#J@WG4<VT$mh !1.@Times New Roman------ 2
n` 0 Spring 20
2
n0 1
2
n0 8 2
nP0 /2
n0 Professor Z. Damnjanovic
2
n0 2
` 0 M, W, F 9
2
0 -
2
0 9
2
0 : 2
0 50 2
0 , 2
0 SOS B37
g2
=0 2
0 (213) 740
2
;0 - 2
A0 4084
2
a0 2
`0 Office hours:
2
0 M
2
0 ,
2
0 2
0 W, F
2
0
2
0 8
2
0 - %2
0 9 R2
I/0
2
0 E
2
0 - ,2
0 mail: zlatan@usc.edu
2
0 #2
`0 & by appointment
2
0
2
`0
2
`0 2
`0 #2
0 2
0 2
0 GESM 160g:
2
;0 =2
A!0 Seminar in Quantitative Reasoning
2
0
2
`0
2
d0 @2
`#0 (2
0 Taming the Infinite +2
e0 : How Math Came to Be
2
0 2
`0 2
0
2
0
2
&`0 )2
8`0
2
80
2
K`0 2
]`c0 The topic of the seminar is the Greeks invention of mathematics as a logical theory, with special 2
p`V0 focus on the evolution of the idea of mathematical infinity, from the Pythagoreans to >2
`"0 Archimedes. This is an area of in Y2
640 terface among mathematics, physics and philosophy.
2
0
2
`0 2
`^0 In developing conceptual tools and computational techniques for solving various practical and 2
`\0 physical problems, mathematicians very early on confronted a fundamental difficulty: how to #2
`0 reduce reasoning
2
0 |2
K0 about continuous, spatially extended magnitudes such as area and volume to 2
`0 numbers 2
0 --
2
0 52
0 aggregates of discrete, atom
2
W0 - ^2
]70 like units that can be counted. Attempts to solve the 2
`V0 problem directly seemed to lead straight into paradox and contradictions. Sophisticat 2
~0 ed 2
``0 strategies evolved to bridge the gap between the continuous and the discrete, the geometric and 2
`O0 the arithmetical, through construction of the new systems of irrational numbers 2
V0 --
2
b0 2
i0 numbers V2
(`20 determinable only by means of an infinite process 2
(0 --
2
(0 2
(0 and infinitesi 2
(0 mal numbers 2
(]0 --
2
(i0 2
(l0 numbers 2
:``0 smaller than any finite positive number yet not equal to zero. Unable to do without them, and 2
L`X0 trying to justify reasoning about them, mathematicians found themselves unwittingly and U2
_`10 unwillingly ensnared in problems logical, philoso P2
_.0 phical, physical and metaphysical. We follow 2
q`_0 the struggle against these difficulties, learning the ever more ingenious methods of reasoning /2
`0 invented to tackle them.
2
0
2
`0
2
d0 2
`X0 No knowledge of any particular branch of mathematics (including calculus) is presupposed 2
0 . 12
`0 Lectures assume only high
2
0 - %2
0 school algebra.
2
y0 @Times New Roman------
2
`0 --- =2
`!0 Tentative list of topics covered:
2
(0
2
`0 @"Arial---------
2
d0 ----
2
i0 --- 2
|R0 early computational mathematics: finding square roots and greatest common divisors
2
0 0'---
2
d0 ----
2
i0 --- _2
|80 Egyptian mathematics: computing areas and volumes by cut
2
0 - 2
0 and
2
0 - 2
0 paste
2
D0 0'---
2
&d0 ----
2
&i0 --- ,2
&|0 the Pythagorean theory
2
&0 2
& 0 of number
2
&S0 0'---
2
8d0 ----
2
8i0 --- %2
8|0 geometric algebra
2
80 0'---
2
Kd0 ----
2
Ki0 --- b2
K|:0 the paradoxes of Zeno: are there atoms of space and time
2
K0 0'---
2
]d0 ----
2
]i0 --- 2
]|]0 the three classical problems: squaring the circle, doubling the cube, and trisecting an angle
2
]0 0'---
2
pd0 ----
2
pi0 --- @2
p|#0 the discovery of incommensurability
2
pb0 0'---
2
d0 ----
2
i0 --- D2
|&0 axiomatization of geometry: why and ho
2
0 w
2
0 0'---
2
d0 ----
2
i0 --- 72
|0 Eudoxean theory of proportion
2
>0 0'
2
d0 - 2
j0 S2
}00 the method of exhaustion: Euclid and Archimedes
2
0 "Systemuxb
uu7
-- 00//..՜.+,D՜.+,hp
vHewlett-Packard Company1a Title Headings 8@_PID_HLINKSAtHn:!http://dps.usc.edu/]http://emergency.usc.edu/6"https://diversity.usc.edu/if$https://studentaffairs.usc.edu/ssa/m9http://dsp.usc.edu/IFAhttps://studentaffairs.usc.edu/bias-assessment-response-support/nwhttp://equity.usc.edu/
http://sarc.usc.edu/SF "https://engemannshc.usc.edu/rsvp/jQhttps://urldefense.proofpoint.com/v2/url?u=http-3A__www.suicidepreventionlifeline.org_&d=DwMFAg&c=clK7kQUTWtAVEOVIgvi0NU5BOUHhpN0H8p7CSfnc_gI&r=CkL4kY0no88QZj2uCDFOI7xhjLQ9O4YZiOIPGjQSQJ8&m=FN8fbuOqtif1bWvrKTHrycKDSgd_snFubXhYrp7Dz58&s=yigxVd9JJfatu9uXYx0qpj6LhjknPiz6tnBwI27pj4U&e=0/'https://engemannshc.usc.edu/counseling&y'https://policy.usc.edu/scampus-part-b/
!"#$%&'()*+,-.0123456789:;<=>?@ABCDEFGHIJKLMOPQRSTUXRoot Entry F@MZ1Table WordDocument4:SummaryInformation(/=DocumentSummaryInformation8NCompObjr
F Microsoft Word 97-2003 Document
MSWordDocWord.Document.89q