Trench Coats: Hot Look for Spring

The thermometer is screaming winter in many parts of the country, but stores are already showcasing clothes for warm weather.So, in The Early Show's "Chic on a Shoestring" Wednesday, InStyle magazine Market Director Toby Tucker Peters offered a sneak peek at a big spring trend: trench coats. They were fashionable last year, too, but this time around, they're out in a wide variety of styles and cuts, so you'll find something right for you.In "Chic on a Shoestring," expensive "splurge" clothing is shown along with much more reasonably priced items, referred to as "steals." It's often hard to tell the difference. Trending News David Ortiz In Boston Jupiter's Moons NYC Helicopter Crash Tick-Borne Diseases Rise LOOK NO. 1: DOUBLE-BREASTED, A CLASSIC STYLE, PERFECT FOR WORK1) Splurge: An Earnest Cut & Sew, $550 at Barney's New York. Crisp, tightly woven fabric helps maintain silhouette. Stretch cotton keeps it neat shape to repel spring showers. A touch of yellow in the blouse or shoes. A hot color for spring. a. Chocolate brown trench coat: Earnest Sewn, $550 (available at Barney's)b. Yellow blouse: Club Monaco, $129c. Capri jeans: 7 For All Mankind, $138 (available at Bloomingdale's)d. Yellow Patent Amina Sandal: Cole Haan, $295e. Vida Gold Vermeil Earrings: Alex Woo, $2182) Steal: Cotton, khaki versatile, belted waist can emphasize or conceal curves worn with a belted cardigan to complete the look.a. Khaki trench coat: Ann Taylor, $188b. Yellow tank top: Urban Outfitters, $12c. Blue cardigan: Eva Franco, $125d. Belt: Urban Outfitters, $28e. Blue wide leg trousers: Theory, price unavailablef. Reed peep toe flats: Gianni Bini, $69What's the difference? The splurge trench is made with a water repellent fabric.LOOK NO. 2: CROPPED: A REFRESHING NEW TAKE ON THE TRENCH COAT. A FUN WAY TO TAP INTO THE TRENCH COAT TREND. 1) Splurge: Pairs well with slim pants or shorts to complement the trim style.a. Dark brown cropped trench: Sportmax, $940b. Shorts, Tocca, price unavailablec. Blouse, Theory, price unavailabled. Shoes, Giani Bini, $69 e. Zahrah Vermeil Ring: Alex Woo Jewelry, $248f. Sienna Vermeil Hoop Earrings: Alex Woo Jewelry, $2382) Steal: Short trench is perky and youthful. Yellow kickstarts any spring wardrobe. Finish the outfit with a pair of peep toe flats - another spring must-have. a. Yellow cropped trench: J. Crew, $168b. Ruffle Shirt in Cherry: 7 For All Mankind, $151 (available at Saks)c. Cropped jeans: Urban Outfitters, $49d. Lana Duet bag: Stuart Weitzman, $450e. Silver and denim peep toe flats: Luciano Padovan, price unavailableWhat's the difference? You're paying for the material in the Sportmax trench.LOOK NO. 3: BRONZE, GOLD AND SILVER TRENCHES ADD TONS OF GLAMOUR TO ANY EVENING LOOK1) Splurge a. Metallic trench: Badgley Mishka, $895 (available at Neiman Marcus)b. Black bubble dress: Signette, $129c. Black patent and gold shoes: Jimmy Choo, $580d. Gold bag: Moë, price unavailablee. Gold hoops with green stones: Anthropologie, $482) Steala. Silver metallic trench: Anne Klein, $495 (available at Nordstrom)b. Black dress: Urban Outfitters, $178c. Natural Python Catarina Sandal: Cole Haan, $325d. Snakeskin clutch: Urban Outfitters, $28e. Necklace: V'ta Jewelry, price unavailableWhat's the difference? The fit and material are what you're paying for with the more expensive trench.

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1.Thingsyoucan'tproveinmathWhatyou'reprobablysearchingforinalongerdiscussionissomethingalongtheselines.Inshorttheansweris:Thereisnosuchthingyetandtherewillnotbeforaverylongtime.Giventhefactthattobuildsuchanon-contradictingconstructiscurrentlybeyondhumancomprehension.ThereasonIsaythatisduetothecomplexityoftheentirefieldandveryfewspantheentirecrosssectionofapplicationofpureandappliedmathematics.Theholygrailofpuremathematicsistofindsuchbeautifulthingsthattieallthistogetherandmakeacoherentsenseoftheentireuniverseofmathematics.Roughlybrokenintothe"fourpillarsofmathematics."Beyondthisyouhavetheideasthattherearemanydifferentkindsofsettheoriesandtheyarecreatedtoworkindifferentworldssotospeak,assuchyou'llfindtherearemanyalgebrasalsofordifferentreasoning.Mostofthesettheoriesfallpainfullyshortofsomesortof"latinofmathlanguage"ifyouwill.Alongwiththefactthatmathematicsisactuallycreatedontheflybynewpeopleeverydaytodescribeourworld.Therehavebeentimesthatphysicsphenomenacouldnotyetevenbeproperlymechanicallyexplaineduntilthemathematicswasinventedtodescribeit.Apersonalsohastocontemplatemathematicsasawaytotakeanun-exactingnaturaluniverseweliveinandtranslatethattoafairlyexactingsetoflanguagethatmathematicsreallyis.Reference:giveatasteofthedepthandcomplexity(thisisnotallencompassingeither.)FoundationsofMathematicsFoundationsofmathematicsisatermsometimesusedforcertainfieldsofmathematics,suchasmathematicallogic,axiomaticsettheory,prooftheory,modeltheory,typetheoryandrecursiontheory.Thesearchforfoundationsofmathematicsisalsoacentralquestionofthephilosophyofmathematics.PhilosophyofMathematicsThephilosophyofmathematicsisthebranchofphilosophythatstudiesthephilosophicalassumptions,foundations,andimplicationsofmathematics.Theaimofthephilosophyofmathematicsistoprovideanaccountofthenatureandmethodologyofmathematicsandtounderstandtheplaceofmathematicsinpeopleslives.Thelogicalandstructuralnatureofmathematicsitselfmakesthisstudybothbroadanduniqueamongitsphilosophicalcounterparts.CommentsonSetTheoryThemomentumofsettheorywassuchthatdebateontheparadoxesdidnotleadtoitsabandonment.TheworkofZermeloin1908andAbrahamFraenkelin1922resultedinthesetofaxiomsZFC,whichbecamethemostcommonlyusedsetofaxiomsforsettheory.TheworkofanalystssuchasHenriLebesguedemonstratedthegreatmathematicalutilityofsettheory,whichhassincebecomewovenintothefabricofmodernmathematics.Settheoryiscommonlyusedasafoundationalsystem,althoughinsomeareaswhich?categorytheoryisthoughttobeapreferredfoundation.Elementarysettheorycanbestudiedinformallyandintuitively,andsocanbetaughtinprimaryschoolsusingVenndiagrams.Theintuitiveapproachtacitlyassumesthatasetmaybeformedfromtheclassofallobjectssatisfyinganyparticulardefiningcondition.Thisassumptiongivesrisetoparadoxes,thesimplestandbestknownofwhichareRussell'sparadoxandtheBurali-Fortiparadox.Axiomaticsettheorywasoriginallydevisedtoridsettheoryofsuchparadoxes.Manymathematicalconceptscanbedefinedpreciselyusingonlysettheoreticconcepts.Forexample,mathematicalstructuresasdiverseasgraphs,manifolds,rings,andvectorspacescanallbedefinedassetssatisfyingvarious(axiomatic)properties.Equivalenceandorderrelationsareubiquitousinmathematics,andthetheoryofmathematicalrelationscanbedescribedinsettheory.Settheoryisalsoapromisingfoundationalsystemformuchofmathematics.SincethepublicationofthefirstvolumeofPrincipiaMathematica,ithasbeenclaimedthatmostorevenallmathematicaltheoremscanbederivedusinganaptlydesignedsetofaxiomsforsettheory,augmentedwithmanydefinitions,usingfirstorsecondorderlogic.Forexample,propertiesofthenaturalandrealnumberscanbederivedwithinsettheory,aseachnumbersystemcanbeidentifiedwithasetofequivalenceclassesunderasuitableequivalencerelationwhosefieldissomeinfiniteset.Settheoryasafoundationformathematicalanalysis,topology,abstractalgebra,anddiscretemathematicsislikewiseuncontroversial;mathematiciansacceptthat(inprinciple)theoremsintheseareascanbederivedfromtherelevantdefinitionsandtheaxiomsofsettheory.Fewfullderivationsofcomplexmathematicaltheoremsfromsettheoryhavebeenformallyverified,however,becausesuchformalderivationsareoftenmuchlongerthanthenaturallanguageproofsmathematicianscommonlypresent.Oneverificationproject,Metamath,includeshuman-written,computerverifiedderivationsofmorethan12,000theoremsstartingfromZFCsettheory,firstorderlogicandpropositionallogic.MathematicsAsTheoryandLogicalReasoningMathematicsisthestudyofquantity,structure,space,andchange.Mathematiciansseekoutpatternsandformulatenewconjectures.Mathematiciansresolvethetruthorfalsityofconjecturesbymathematicalproof.Theresearchrequiredtosolvemathematicalproblemscantakeyearsorevencenturiesofsustainedinquiry.SincethepioneeringworkofGiuseppePeano(1858-1932),DavidHilbert(1862-1943),andothersonaxiomaticsystemsinthelate19thcentury,ithasbecomecustomarytoviewmathematicalresearchasestablishingtruthbyrigorousdeductionfromappropriatelychosenaxiomsanddefinitions.Whenthosemathematicalstructuresaregoodmodelsofrealphenomena,thenmathematicalreasoningoftenprovidesinsightorpredictions.Throughtheuseofabstractionandlogicalreasoning,mathematicsdevelopedfromcounting,calculation,measurement,andthesystematicstudyoftheshapesandmotionsofphysicalobjects.Practicalmathematicshasbeenahumanactivityforasfarbackaswrittenrecordsexist.RigorousargumentsfirstappearedinGreekmathematics,mostnotablyinEuclidsElements.MathematicsdevelopedatarelativelyslowpaceuntiltheRenaissance,whenmathematicalinnovationsinteractingwithnewscientificdiscoveriesledtoarapidincreaseintherateofmathematicaldiscoverythathascontinuedtothepresentday.PerhapsNotGalileoGalilei(1564-1642)said,Theuniversecannotbereaduntilwehavelearnedthelanguageandbecomefamiliarwiththecharactersinwhichitiswritten.Itiswritteninmathematicallanguage,andthelettersaretriangles,circlesandothergeometricalfigures,withoutwhichmeansitishumanlyimpossibletocomprehendasingleword.Withoutthese,oneiswanderingaboutinadarklabyrinth.CarlFriedrichGauss(1777-1855)referredtomathematicsastheQueenoftheSciences.BenjaminPeirce(1809-1880)calledmathematicsthesciencethatdrawsnecessaryconclusions.DavidHilbertsaidofmathematics:Wearenotspeakinghereofarbitrarinessinanysense.Mathematicsisnotlikeagamewhosetasksaredeterminedbyarbitrarilystipulatedrules.Rather,itisaconceptualsystempossessinginternalnecessitythatcanonlybesoandbynomeansotherwise.AlbertEinstein(1879-1955)statedthatasfarasthelawsofmathematicsrefertoreality,theyarenotcertain;andasfarastheyarecertain,theydonotrefertoreality
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