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Sample translations submitted: 1
German to English: Das Mathematische im Menschen
Source text - German 9999999Das folgende konnte auch eine Einleitung zu einer Geschichte der Mathematik sein. Es wird der Versuch gemacht, die Mathematische Wissenschaft mit der Gesamtheit der Menschlichen Betätigung in Verbindung zu bringen, wobei die Enge des zu Verfügung stehenden Raumes an vielen Stellen eine Beschränkung auf Andeutung notwendig macht. Zuerst (1-10) stellen wir die Wurzeln der Mathematik in der menschlichen Seele dar. Dann (11 ff) zeigen wir, wie in der geschichtlichen Entwicklung der Mathematik das allgemein Menschliche zum Vorschein kommt. Unsere Betrachtungen haben vielleicht einen gewissen Wert für die so wichtige Frage nach der Bedeutung der Ordnung für den menschlichen Geist. An manchen Stellen werden wir auch das Problem der Entstehung und Wirkung von Kunstwerken streifen, ohne natürlich mehr als eine ganz beschränkte Seite dieses Problems zu betrachten.
Translation - English Das Mathematische im Menschen (The Mathematical in Man)
The following could also be the introduction to a history of mathematics. The attempt will be made to connect the study of mathematics to the totality of human operations, whereby the narrowness of the available realm we have at our disposal makes it necessary at many points to limit ourselves to simple suggestion. First (1-10), we will illustrate the roots of mathematics in the human soul. Then (11 ff) we will show how the generally human emerges in the historical development of mathematics. Our considerations have perhaps a certain worth for the very important question of the meaning of order for the human spirit. In many places we will also touch on the problem of the emergence and effect of art works, without of course considering more than a very limited side of this problem.
1. Under rhythm we understand in the following the succession and accentuation of things which are the same, or which change according to a rule. We want to attempt to show how the occupation of the rhythm-creating person is linked with the occupation of the mathematician. We will, with our considerations, differentiate various kinds of rhythms:
One-dimensional rhythm (simple music). The linear, one-dimensional rhythm expresses itself somewhat in the drum beats which repeat themselves, weakly and strongly, in the same intervals, in strict time and the dance which corresponds to it. When, as the ethnologists report, in particular tribes several different rhythms and drumbeats are performed simultaneously by different persons, there occurs already a situation which is close to a mathematical operation. The pleasure in the regular recurrence of the moments where the emphasized beats, the downbeats of the various rhythms concur, matches the pleasure of the number theorist in his first beginnings, when he observes how the sets associated with the multiples of various numbers are embedded in each other. Both emotions spring from the same basic characteristic of the human soul.
Moving from this most simple musical operation to more complex structures, we come perhaps to the imitation where the various themes in the various successive voices become intertwined with one another.
Since time immemorial the discourse has been on the connection of music with mathematics. But our observations are not to be confused with mathematical descriptions of acoustic conditions. The Pythagoreans, for example, linked the physiological, namely the sound relationships, with the length of the strings. That is essentially distinct from what is being considered here. Nevertheless they may have had a certain idea of the common source of the musical and geometrical emotion shown here.
2. Multidimensional rhythm (Architecture, ornamentation, symmetry, conversion). One can naturally regard a joint as a multidimensional rhythm, even more so for that of the complex general works of music, about whose effect we will later say quite a bit more. However the one dimension, time, is always so strongly emphasized that for the consideration of multidimensional rhythm, we would rather go into the realm of the visual. Here we have, in our sense, a particularly simple situation in the consideration of the façade of a large, gothic church, such as that of the Strasbourg Cathedral or the Cathedral of Notre Dame in Paris. Rectangular, horizontal stripes repeat themselves in various sizes. These stripes are divided again, so that they show simple symmetries along vertical axes. Here one cannot suppress the fact that these simple tapestries of sound receive awaken a feeling through their sizes which so multiplicatively exceed the human measurements: this feeling arises from a part of the soul quite distant from the mathematical sensibility. The very simple mathematical structure of the pyramid is likewise quite essential. But what would the pyramid be without its tremendous dimensions, with which the beholder also perceives the astounding dimension of the space of time since its erection and finally, compelled by the firmness of the material, its long endurance in the future. The size-ratios also play a particularly important role in architecture, but nor do they immediately arouse the mathematical sensibility. An entirely different question is whether perhaps in the construction of some structures, intricate mathematical relationships have come into consideration. For these works as art objects such relationships are only of importance when they can function directly. They have, in any case, nothing to do with the mathematical dealt with here.
3. We easily find an example for the occurrence of three-dimensional rhythms in the organization of the interiors of large churches. We need only to think of the arrangement in the aisles or of the organization of the supporting parts of the arch in rows of pillars and galleries which run one on top of the other, etc.
The rhythm in the various forms is realized particularly strongly in the ornamental. We consider, for example, the quite primitive ornaments on a pottery vessel from the Stone Age: rows of notches placed under each other, alternating rows of stripes that incline to the left and to the right, etc. Already in the consideration of architectural rhythms we’ve found another relationship important to mathematics: symmetry, a particular kind of spatial relation. Here in the ornamental appear an additional two things important for mathematics: first, the appearance of that which continually alters, but in its alteration always repeats itself, namely geometric shapes with an endless group of transformations in themselves. This corresponds to the periodical functions or, more generally, the unlimited repeatability of the same process. The simplest figure of its kind, the wave line, is present in the ornamental, then come more complex figures such as the Greek fret.
Moving to two-dimensional ornaments, to mosaics, for instance, we encounter surfaces that, always repeated, can be placed on top of each other without mutually disrupting each other on the unbroken filling of the levels with congruent figures, likewise a special case of the unlimited repeatability of an algorithm.
Thus we see here an initial interplay of fantasy with the attempt recognize an external constant. First, certainly, quite unconsciously, as with the pottery vessels from the Stone Age, where occasionally a sudden shift in the ornamentation occurs along with the kind of ornamented surface. These vessels have, namely, a saddle-shaped (negatively) curved upper piece, which, on the horizontal circle, which consists of the inflection points of the profile lines, transitions into a positively curved piece. At this point the types of ornamental band change, often without transition. We find this conversion, in mathematical language the solution of a boundary value problem, often also in architecture. We think, perhaps, of the stairs in in the style of the natural terrain, and our geometrician’s heart is thrilled when we imagine again the magnificently ascending flights of stairs in the interior of the Genovese Palaces. The ornamental plays, with its problems, a very large role in sculpture as well. We consider, for example, the hair on the archaic Greek heads. These are broken up, perhaps also in geometrically distinct zones, which are imbued with similar forms, strands of hair, locks, bulges, and waves. We find the like also in the wings of the Hittite sphinxes and in fact everywhere where we are situated in the area of the so-called archaic art.
4. A tremendous realm of mathematics in art presents itself with the principal of the succession of shapes which uniformly change. We wish to attempt to describe this high degree of rhythm with an example. We select the ornament on a board from New Guinea. On the longitudinal axis the middle points of ovals symmetrical to the axis lay in the same intervals. The interior of the ovals are each filled out by a spiral. After the edge the ovals are followed by curved lines. The curved line which touches the edge builds with this a row of closed curves, the interior of which is filled out again by closed curves. Here we have the completed example of a curve display field before us, which has certain constraints to satisfy. Without having such pictures before one’s eyes, the analyst will hardly be able to solve functions by which such systems of curves are exactly determined by metric qualities. It is quite delightful to observe how directly the artistries in the various regions differentiate themselves in the manipulation of this principle. In Central Australia, for example, we see manifold attempts to get away from the usual ornamentation. However the goal doesn’t seem to be reached. On the other hand we find in the neighboring New Guinea innumerable examples of the higher ornamentation, which indeed exist alongside the most beautiful achievements of civilized peoples. Here we naturally find also the spiral ornament, which can certainly be considered characteristic for the transition to the higher degree. This transition parallels the transition in mathematics from the arithmetic to the functional.
With this conception, we now enter the territory of great art. We see the charming cascade of lines on the clothing of the “Ludovisi Throne,” the three-dimensional folds on the Nike figures which give the powerful impression of movement, and finally the entirely three-dimensional solution to the hair problem, [that is] the locks and strands of hair which push into and weave over and through each other and smooth themselves out from the passionately heightened excitation to the peacefulness of relaxation. The mathematician must profess that his feeling for the richness, which the simply viewable transcendent appearance of the room experiences a dreamlike intensification in the consideration of such works of the highest Greek art.
We experience the like through music. Here, too, the succession of things which change according to a law is something quite natural. Melody, accompaniment, modulation and polyphony offer the possibility of the many-dimensional. Hereby we are versed in differentiating intuitively the style of the various composers, but also the objective analysis is possible leads the appraiser, exactly as in the visual arts, to reliable statements about the origin of artworks.
5. – Thus far we have recognized in sources of creativity: the inner desire for rhythm in the broadest sense, as well as the delight in wealth and abundance, meaning here in the multiplicity of connections of multidimensional shapes, finally the necessity to conform to the given circumstances. Particular to the visual arts, the drive to representation and reproduction is another important source. Many wish to view this drive as the first source of every artistic pursuit. Surely the delight in reproduction is a quite primitive feeling, as well as the urge to permanently hold on to the fleeting occurrence. The sphere of the mathematical has perhaps a share even in both, though artistic creativity is hereby only stimulated. The real power comes from the original emotions, from which we have once more summarized that which we have considered here.
It is something quite different that from time to time, the study of mathematics gives aid in the representation. We will come back to this later when we deal with the topic of applied mathematics.
6. – There is something different which is closely connected with representation and reproduction: the symbolic. In the artistic operations as well, the symbolic, as well as the replacement of the complex by something more simple, and the concretization of the abstract, plays a large role. Although there is hardly anything in which the spirit of man is engaged that would not essentially contain these processes. Here the area has gotten therefore so large that we must cease with general considerations so as not to go beyond the narrow boundaries of this article. We have reached the point where the mathematical sense-complex passes into the most inward and most important central organs of the human spirit. In the overview of the history of mathematical studies, we will emphasize the meaning of this process.
7. – A special case of symbolization is the spatial illustration of operations. Today we find them everywhere in daily life, perhaps in the simplest form in family trees or schematics for the subordination in complex organization, moreover in the manifold “graphic representations” which today, observably and impressively, effortlessly shape the collection everywhere of direct of numerical figures through the comprehensive picture. In some of these pictures, some small mathematical facts are already visible. But here we are still very far removed from the study of mathematics. We come somewhat closer to it when we observe the discovery of operations which follow a set pattern. This way we can calculate the discovery of the simplest number relations, or the perception of topological qualities of even curve systems, which is excited by the rhythm of the second stage, moreover the discovery and illustration of simple operations in physical processes, the discovery of “models” for the physical phenomena.
8. – In the first sections, we spoke always of the emotional, somewhat of the delight in the cluttering together of several rhythms. Last, we spoke of the conscious perception, for instance, the curious operations between several rows of multiples of numbers. However, we are still not in the kingdom of the study of mathematics. The transition to that which we designate the study of mathematics or, in short, as mathematics, occurs now with the appearance of one of the only other pursuits which is characteristic for mathematics. This pursuit alone makes from everything that which is created through mathematical emotions, that which is discovered in the mathematical, the uniform figure of the study of mathematics, which has developed in such an astounding way for four thousand years. Yes, a single figure. Because unlike with artworks, with collections of facts, here stand innumerable results of work next to each other, each with its own independent meaning. Rather, all are connected through that which is characteristic for mathematics, the framework of logical conclusions, whose foundation rests upon a few, simple conditions. Through the conscious or unconscious applied methods of the logical conclusion, the “producing’ of hitherto unknown facts, mathematics emerges.
We can regard these logical conclusions as a kid of inner ornamentation which, divided in condition, proposition and proof, is first illustrated by the Greeks. However previously, there existed already a study of mathematics. The inner ornamentation was simply not projected to the outside, so that today we see only the results, absent their derivation. The mathematicians are the ornamentalists in the logical conclusion. The rules of this ornamentation are very strict, and it is therefore astounding, how the tremendous work of art can nevertheless arise in that on which all mathematicians from the time of the Babylonians have worked.
9. – After we have ourselves entered the real of mathematics, we must, before we speak of its historical development, consider and critically examine a divide of this realm, namely the divide between pure and applied mathematics. Everything that has so far appeared in mathematics is, taken in the strictest sense, applied mathematics. Because pure mathematics derives newly from given operations without consideration as to whether these operations occur perhaps in the numbers or solid figures through suitable interpretation. In this sense, arithmetic and geometry, indeed itself the most abstract area of mathematics, group theory, therefore able to be designated as applied mathematics. It is indeed however a substantial difference between them and that which one normally refers to as applied mathematics, and what should be so referred to hear as well. In geometry, arithmetic, etc. everything is constructed, through logical structures, on a few basic premises, whereby the consideration for what is important geometrically or arithmetically certainly always determines the direction of the deduction. In the fields in which mathematics is applied, operations are discovered. These are brought into the common mathematical language, out of them conclusions are mathematically drawn, and they are finally interpreted again in the original field. The logical-mathematical constructions are invariably only intermediate stages; the “axiomatic” method is only feasible in pure mathematics.
Shortly here we want to refer to the meaning of applied mathematics on both sides. One can give a vivid description of spatial figures through perspective illustration. Should a painter wish to use this method, in this way a familiarity with the simplest mathematical qualities of perspective illustration is useful. So arrived Leonardo da Vinci at the saying that a background in mathematics is essential to painting and an understanding of the painting. On the other hand, we owe the first developments in the projective geometry through Desargues to the interest of the Renaissance painters. Problems of the mechanics of the continua led at the end of the 18th century to the most important problems of analysis. The theory of relativity used already available geometrical figures to this end.
10. – The mathematical method always makes a great impression. One would gladly like to deal with all that is possible “more geometrico.” The attempt of Spinoza’s to axiomatize ethics had to fail. However even in the national economy, where number relations doubtlessly play a large role, the mathematical method has only been able to be used in a quite primitive form.
In the occult sciences, the drive to dialectically derived insights expresses itself quite curiously, where incidental connections prompt more foundational connections, which, again with a pseudo-dialectic, lead to further connections, namely the sought-after derived insights, to be fixed. In the realm of such an occult science, the study of mathematics can also have relevancy, as in astrology. Here the mathematically-governed movements of the stars is mapped onto the courses and linkages of the destinies of the human world, since we would so gladly like to know something of the future which lies in darkness, yes, indeed, as it propels us from the deep, that which appears lawless, to rule through the recognized law. Other occult sciences have, to be sure, nothing to do with the study of mathematics, though indeed with mathematics in a more general sense. In this way chiromancy, where the rhythm of the second phase, which furnishes the nature in the curve system on the palm of the hand, prompts the beholder to conclusions on the course of the destiny in past and future. Nowadays the connection of mathematics with the occult sciences is strange or troublesome to us. In the first centuries after Christ, one understood exactly such occult scientists under “mathematici.” And Kepler as well was famously compelled professionally to astrology. His inclination certainly also corresponded to the other drive cited above to govern the seemingly random through mathematical images, to discover the “harmonia mundi.”
11. – If we make the attempt now to illustrate how the enormous blanket of mathematics came into being in the course of time, we must use a somewhat amplified approach. It is not our aim to give an encyclopedic overview of mathematics, but rather we want to experience how this science is connected with the people who produce it. To this end, we have so far considered the mental faculties of the individual. Now, in the historical description, we must necessarily take into consideration the milieu in which the science arose. In actuality, nowhere in the whole development, even when it is defined by individuals, is the milieu without importance. So we must imagine the heretofore oldest known documents of mathematics (Babylonian clay tablets from the beginning of the second century) as arising in a community. From these tablets it emerges, that the ancient Babylonians knew the relationships between the sides of a right triangle, that is the Pythagorean Theorem, in algebraic form, and as far as calculating, understood that they could apply this knowledge to geometric computations. We are particularly interested in the form in which these things were stated. Here we have only to do with special problems, whose solution is presented every time without further explanation. The enclosed figures will not be operated on. They serve as the basis for concise descriptions of the problems. These are themselves altogether only without actual practical meaning, but only in part also without any geometric meaning, since measures are assumed which are not connected geometrically. Such problems are doubtless there in order to provide manifold examples for quadratic equations. They are thoroughly consistent with a certain kind of exercises that are still in use today. Already the word “problem” leads us into the milieu of origin. The clay tablets belong to a school, where the results of the research appeared in exercises.
Puzzling is the origin of the indeed astoundingly far-reaching results, for which we can consider the available floor plan drawings as a precursor at most. (No evidence exists, that the Pythagorean Theorem was known to the ancient Babylonians in geometric form. Between this and the algebraic form is indeed also a very large gap for the beginners). We know just as little about the milieus of origin of Babylonian mathematics. One can imagine a kind of shed, perhaps under the management of priests, who undertook great structures; we are indeed used to concentrating everything spiritual of the ancient peoples of the Middle East in the priestly communities, compelled surely by Greek accounts.
In summary we must say, that from the oldest mathematics only the curiously well-developed results are obvious; of their origins, however, nothing is surviving. In this only discoveries and decipherments can assist further. From that which is surviving we gather that the Babylonians had an interest, perhaps arising from the applications, in the computations of geometric sizes, but that algebra was practiced without such concerns, out of pleasure in calculation, almost in the complexity that it had up until the Renaissance.
For brevity’s sake, we don’t want to elaborate on the Egyptian mathematics. From the transmission in two papyrus manuscripts it seems to follow that the ability there in calculating, in operation with symbols, and the pleasure in such things as well was very much lower than for the Babylonians. A curious singular instance the surviving formula for the truncated pyramid from the middle of the second century.
12. – The Greeks have for their posthumous reputation the incomparable kindness of fate that not only their life and their deeds survive from great poets in a form that is linguistically, aesthetically and ethically near to us, but also that we have from their thinkers essentially exact messages and that many of their chief works survive without gaps. Precisely with the Greeks begins the continuity of our Western civilization. Perhaps under the reasons for this continuity one may point out that the Greek poets, artists and philosophers gave their works a quite clear, completed form, which consistently functions as an indestructible, consummate whole. And so they brought the mathematical results, through the logical strutting, into an indestructible form, in a classically clear form through the structured construction from foundation up until the highest achieved.
We first want to speak about the social milieu in which the Greek mathematics arose: research and teaching were closely connected. The researchers emerged almost always emerged in school associations, often under a supreme leader. The newly discovered was relayed to the students and colleagues first predominantly orally, then later often only in writing. In the Platonic Academy in Athens we have a particularly firmly organized form of school in which, intermittently, mainly mathematics was practiced. Such a school contains as well that which parallels the college of an English university and the learned society (academy). At the platonic and similar schools, the religious had an importance which regulated school life. At the Pythagorean School, which preceded the platonic and which, in its esoteric character, originally followed non-Greek institutions, mysticism and the mystical affinity played a large role.
The oldest written description of a mathematical investigation, from Hippocrates of Chios, from the mid-fifth century only survives to us only in obviously revised form from a third party. We know, however, that the same Hippocrates first wrote a book on the “elements,” a cohesive, constructive description in which the proofs are already explicitly described. The logical construction of the doctrinal system is at the time, at least in principle, already in place. This results also from a simultaneous discovery which without such a construction is not at all conceivable, from the Pythagorean discovery of the irrationals. The proposition that sides and diagonals of a square possess no shared measurement is not able to be recognized experimentally and cannot be at all understood without extensive, purely logical proof. In this way with the date ~450 we can determine fairly well the point in time in which mathematics had taken a tremendous step of serious consequence. This step parallels perhaps the unathoritative spirit of the ancient Greek thinkers, as it is clearly expressed, for example, with the Ionic philosophers. The step was of consequence at once for the entire development of humanity, as man received through the strict closing, the strongest weapon for the command of the forces of nature. On the other hand it had immediately deciding consequences for the development of mathematics. When one recognized that the geometric relations [proportions?] cannot be described through relationships between whole numbers, one abandons the calculation which is felt to be insufficient and abandons absolutely operating with symbols and describes everything, indeed the purely number-theoretical, in geometrical form. One geometrizes therefore the non-geometric as well. That is to be understood here as a process opposed to “symbolizing.” Of course the discovery of the irrationals is note the only reason for this development. One may assume that the Greeks were by a considerable measure visual, so that this process of visualization was very attractive to them. Of importance was surely also the influence of Plato. It was platonic to reject crude calculation, which used symbols of its own accord, as philistine. Plato’s influence likely achieved the fact that the elements of Euclid, roughly 300, were so strictly established that first in the second half of the nineteenth century, amendable places were recognized in this structure. Here the teaching of the irrational relations is with great austerity particularly abstractly developed, but not carried out as far as the derivation of all of the algorithms. To be sure, the logical constructions in mathematics were also influenced by Aristotelian logic.
13. – As we address the history of mathematics, next comes the quite natural process, whereby algebra lost once more the geometric form, into consideration. The first stage here is the appearance of the later Greek Heron in whose works several almost Babylonian-seeming problems occur. Perhaps the Babylonian had always been extant, but did not surface in the official scholarly production.
The mathematics of the Arabs continued the process. In the “Algebra” of Ibn Musa (800), the geometric is present almost only to denote sizes, as the letter codes of the sizes had not yet been invented. In the middle ages the Indians had also practiced mathematics and had likewise facilitated this process of de-geometrizing. The Western world had only one mathematician that we must consider here: Leonardo Pisano (1200). He was not only the editor of “Algebra,” he was also the first who introduced new ideas here. He departed first from the impossibility proofs of the Greeks, whereby he demonstrated a particular equation of the third degree, that its roots are not described by any irrationalities addressed by Euclid. In this the way leads from the Pythagoreans over Leonardo to Abel and Galois, without leaving the Western world.
14. – It is very difficult to describe the manifold development of mathematics since the Renaissance in the linear world of language. It is as much also of importance for our considerations; in general things: social milieu, influence of philosophy, the importance of great mathematical discoveries which continue to have an effect, other sciences; in the particular: above all the struggle of the symbolic with the descriptive in the broadest sense.
We begin with an arrangement of the development from the Renaissance until now. The first part of the mathematical research continues the old research and erects an enormous edifice, whose structure is always able to be newly undertaken when the magnitude of details threatens to suffocate the research. The second part conquers new territories for pure mathematics. One of these territories is logic. It is the hope that from the connections of logic, a truly fertile structure, rich in theory and theorems, is to be established. Another territory is mathematics itself. The theorems of geometry for instance are intertwined through the fact that from certain of these theorems, others which are purely logical can be inferred. These connections are studied. The first very important result of this is the theorem of the unprovability of the parallel axiom due to the “remaining” [odd?] axioms. In geometry as in algebra these studies were successful.
And now a bit about the symbolic. At the end of the fifteenth century one sees with Chuquet that the symbolic takes on a life of its own. The algorithms for irrationalities are discovered (not proven), reckless calculations with imaginary expressions are also made. Calculation with the imaginaries is attended as well, through Cavalieri and Leibniz, by calculation with “infinitely small” sizes. The success decides on the correctness of the procedure. Mathematics had disengaged itself as it were from the vexatious puritanism of Plato. The technique of algebraic calculation is infinitely simplified by the so far overlooked methods to denote all occurring sizes and connections with symbols. First in the nineteenth century, symbolism is again closely connected with visualization. With it, however, it experiences the extraordinary good fortune in set theory of “the free creation of the human spirit,” which, however, inasmuch as it leaves absolutely the ground of visualization, only a modest existence alongside other extensions of mathematics is preserved.
Since Descartes the symbolic or algebraic dominates geometry in exact contrast to the circumstances of the Greeks. This corresponds to the non-Greek notion that all geometric figures are nothing more than point sets. Through Desargues the concept of infinitely extending lines was introduced as a primitive notion that suggests the algebraic approach. Analytical geometry, as it was clearly described at the end of the eighteenth century, brought the notion of the three-dimensional unlimited space as a manifold of Pythagorean triples which is so familiar to us. Projective geometry, especially the teaching of duality in space and plane, delivers the geometric figures from bondage through the point. They became once more autonomous spatial elements.
The beginning of this development lies in the dynamic Renaissance. The fruitful unrest, the revolutionary energy, the first jubilant successes (resolution of the equation of the third degree) provided the beginning. The general spiritual interest, the collaboration in small and large lattices, the successes which always stimulate new work in the pure and applied sciences, carried on.
The historical and psychological consideration provides a certain reassurance, but it will not cripple the research Through the consideration of that which is gone by we are able to free ourselves from the flux of the time and to contemplate new goals whose accomplishment will perhaps first be determined in later generations. Through such general considerations, however, we will be more keenly aware of how the mathematicians are woven into the generally human. And foremost, in order to strengthen and perhaps for many, to first awaken this awareness is the above written.
Frankfurt am Main University
M. Dehn
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I am a recent graduate of Smith College in Northampton, Massachussets with a Bachelor of Arts Cum Laude in German Studies and History with Highest Honors in History. Having spent the 2013-2014 academic year studying at the University of Hamburg, I developed and sharpened my oral and written German to great effect.
I have valuable translation experiences translating advanced texts from German into English for professors at Smith (see resume), a number of which I hope will soon be published, and in translating materials for my honors thesis, the primary and secondary source materials for which were almost all in German. I also have experience teaching and tutoring students in German, and writing in that language at an advanced level, and am often called upon by my German friends to assist with texts they need translated into English.
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